Calculus

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1. Set Theory and Functions

1.1 Generalized Distributive and De Morgan’s Laws

Consider a family of sets , with , where is an index set.

Distributive Law:

De Morgan’s Laws:

Proof of the Distributive Law:

Use (1) as example:

Let's prove the first equation.

Proof of De Morgan’s Law:

Use (3) as example:

Let's prove the third equation.

1.2 Fundamental Properties

1.2.1 Spare Rational Number

Consider the set of rational numbers on the interval

Let’s take a value . The points are in intervals with an interval length of . The total length of the union of these intervals is

on leaves at least a length of empty space. We can take to be arbitrarily small. So, the real line is filled with irrational numbers everywhere.

1.2.2 Upper and Lower Bounds of

Since is always false, then the proposition is always true. This means the upper bound of is any real number. By convention:

1.2.3 Countability

Definition: A set is countable if its elements can be arranged into a sequence according to a certain rule.

Every finite set is countable, but not every infinite set is countable.

Consider , then is countable. (Diagonalization Principle) We can arrange them in a matrix-like form: (Remove duplicate elements.)

1.2.4 Cartesian Product

Definition: For any element in set and any element in set , a corresponding ordered pair is formed, and the set of all such pairs is called the Cartesian product of and , denoted as .

1.2.5 Functions

A function is defined by its rule: .

The conditions for a function are:

  1. Domain

  2. Codomain

  3. Uniqueness

Injective (One-to-one): No two values map to one.

Surjective (Onto): Every element is mapped.

Bijective (One-to-one correspondence): Both injective and surjective.

Proposition for injectivity: .

Note: Each element in the domain must map to a unique image, but a unique image does not require a unique pre-image.

Boundedness: A function is bounded if there exists such that for all , . is an upper bound of . is a lower bound of . is an upper bound. is a lower bound.

Monotonicity: Strictly monotonic Monotonic.

Periodicity, Parity, …

2. Sequence

2.1 Continuity of Real Number

Natural numbers are defined by the following 5 axioms (Peano Axioms):

  • 1 is a natural number.

  • Every natural number has a successor.

  • 1 is not the successor of any natural number.

  • The successor is unique.

  • (Principle of Mathematical Induction) If is a proposition about natural numbers such that

Natural numbers (closed to plus and multiply) are extended to integers () by subtraction, and extended to rational numbers () by division.

2.1.1 Upper Bounds and Lower Bounds

Definition: Let be a non-empty set. If s.t. is an upper bound of . If s.t. is a lower bound of .

Upper bound: Least upper bound sup .
Lower bound: Greatest lower bound inf .

Properties of the least upper bound (sup): (Let )

  • is an upper bound of . .
  • Any number smaller than is not an upper bound. .

2.1.2 Supre- /Infimum Existence Theorem

Theorem: Every non-empty set of real numbers that is bounded above has a least upper bound (supremum). Every non-empty set of real numbers that is bounded below has a greatest lower bound (infimum).

Pf.For all ,

Here, denotes the integer part, and denotes the fractional part.

Let

Then any set of real numbers can be represented by definite infinite decimals:

Assume is bounded above. Let be the largest integer among the integer parts of the elements in . Then define: Let be the largest first decimal digit among the elements in . Then define:

…

This implies:

Let . We now prove that is the supremum of .

Step 1: Prove is an upper bound of .

If , then exactly one of the following two cases holds:

  • (*) There exists such that
  • (**) For all ,

If case (*) holds, then .

If case (**) holds, then by comparing digit by digit, we get .

Step 2: Prove that for every , there exists such that .

For any , there exists such that .

Take . Then:

Therefore:

That is, .

Theorem: The supremum and infimum of a non-empty bounded set of real numbers are unique.

2.2 Sequence Limits and Infinitesimals

2.2.1 Definition

Let be a given sequence, . If , such that , . Denoted as

2.2.2 Properties

  • Uniqueness: The limit of a convergent sequence must be unique

Pf.: Take

When , we have and , which is a contradiction.
  • Boundedness: A convergent sequence must be bounded

Pf.: Assume converges. Take , such that , , then

Take Clearly (discussion for all terms)

Note: A bounded sequence may not necessarily converge.

  • Order Preservation: If , both converge, , , and , then ,

Pf.:

Take , by Then Similarly, by Then we have

Def: Sequence with limit equals to 0 is called a infinitesimal.

2.3 Arthematic Operations of Limit

2.3.1 Pre-conditions

.

These introduce two relations:

We can also say and are infinitesimals, for their limit is 0.

2.3.2 Summation

Pf.

2.3.3 Production

Pf.

2.3.4 Division

Pf.

Take such that , . That is, Thus , . Then Let That is,

2.3.5 Squeeze Theorem

If satisfy for , and , then .

Pf.:

Let , then for , , . Therefore That is, ,

2.4 Infinite Sequences

2.4.1 Definition

A sequence is called an infinite sequence if , such that , If , it is called a positive infinite sequence: If , it is called a negative infinite sequence:

Note: Infinite sequence ≠ Unbounded sequence. Counterexample: 1,1,2,1,3,1,4,1,…

2.4.2

Theorem: If , then is an infinite sequence if and only if is an infinitesimal sequence.

Pf.

(i) If is infinite is infinitesimal

That is, , hence:

(ii) If is infinitesimal is infinite

That is, , hence is infinite.

2.4.3

Theorem: If is infinite and for , (uniformly positive lower bound), then is also infinite.

Pf.

Let . Then for , we have

Corollary: If is infinite and , then is infinite.

2.4.4 Arithmetic Operations of Infinite Sequences

2.4.5 Stolz Theorem

Definition: If satisfies , , we call monotonically increasing.

If , it is strictly increasing.

Stolz Theorem: Let be a strictly increasing positive infinite sequence, and

Then

Pf.Case 1:

Let , When : Similarly: Thus: Since is positive infinite: Let , then : That is,

Case 2:

Let , then : Since is infinite: Let , then :

Case 3: (Proof omitted, similar to Case 2)

e.g. 1: If , then

e.g. 2:

2.5 Convergence Criteria (Real Number Continuity Theorem)

2.5.1 Monotone Convergence Theorem

A monotone bounded sequence must converge.

Pf.Assume is monotonically increasing and bounded above, hence has a supremum.

Let , we prove . By definition of supremum: For , and , hence ,

e.g.: Nested Radical Limit

Prove the existence of (n nested radicals)

Let , then ,

(i) Boundedness: Prove ,

, is obvious. Assume , then , holds.

(ii) Monotonicity: Prove

Assume Therefore is monotone bounded and convergent

2.5.2 Fibonacci Sequence Growth Rate

Fibonacci sequence: , . Let , find If , then ; if , then . Since Thus By Monotone Convergence Theorem, and exist Similarly, Therefore

2.5.3 The Number **

Pf.: is increasing, is decreasing, and both have the same limit:

Therefore both sequences are monotone bounded

2.5.4 p-Series Convergence

Let , , discuss convergence of

Case 1:

Let , Since , converges

Case 2:

2.5.5 Harmonic Series and Euler’s Constant**

Let , prove converges. Known: , . Taking logarithms: This indicates Hence monotonicly decays. Let (Euler's constant)

2.5.6 Nested Interval Theorem

Definition: A sequence of closed intervals satisfying two conditions:

Nested Interval Theorem: If is a nested sequence of closed intervals, then there exists a unique belonging to every , and

Pf.By the definition of nested intervals,

That is, and is bounded above by ; and is bounded below by Therefore both and have limits Let By the Monotone Convergence Theorem and uniqueness of limits, Hence If there exists another satisfying Then by the Squeeze Theorem,

e.g.: Prove is uncountable.

Method 1: Nested Interval Theorem

Proof by contradiction. Assume is countable Let Take such that Trisect into three intervals: (Trisection ensures does not belong to at least one interval) Choose one interval that doesn't contain , call it Repeat this process to obtain This gives us where:
  • This is a nested sequence of closed intervals
  • ,
By the Nested Interval Theorem, such that , Therefore , , i.e., but , contradiction

Method 2: Diagonal Argument

It suffices to prove is uncountable Proof by contradiction. Assume it's countable Let the elements be: Construct where , Then , Therefore is uncountable

2.7 Subsequence

2.7.1 Definition

Suppose is a sequence, and is a strictly increasing sequence of positive integers, then is called a subsequence of , denoted as . Obviously, we know if , then

Theorem: If converges to , then any subsequence of also converges to .

Pf., there exists , such that . Since , thus . Therefore, .

Corollary: If has two subsequences converging to different limits, then diverges. (Contrapositive of the above theorem)

Theorem: converges to if and only if every subsequence of has a subsequence that converges to .

Pf.

Sufficiency is obvious.

Necessity: By contradiction, assume that does not converge to .

That is, , such that , there exists with .

, such that . , such that . (taking as ) , such that . , , .

Any subsequence of does not converge to .

2.7.2 Bolzano-Weierstrass Theorem.

A bounded sequence must have a convergent subsequence.

Pf.
Suppose is bounded, i.e., , such that , . Bisect the interval into: , . At least one of them contains infinitely many , denote it as .

Repeat this process to obtain a nested sequence of closed intervals , where each contains infinitely many .

Take .

Take , with .

Take , with .

…

We obtain a subsequence satisfying . By the Nested Interval Theorem, , such that . By the Squeeze Theorem, .

2.7.3

Theorem. If is unbounded, then there exists a subsequence , such that .

Take , such that .

Take , such that .

…

We obtain a subsequence .

is an infinite sequence.

2.7.4 Theorem

If is an infinite sequence, then there exists a subsequence that is also an infinite sequence.

Take , such that

Take , such that

…

We obtain a subsequence .

is an infinite sequence.

2.8 Cauchy Convergence Principle

2.8.1 Cauchy Sequence

Definition (Cauchy Sequence):

e.g.:

Pf.

So

Take , then

e.g.:

Pf.

If we take , then no such exists

2.8.2 Cauchy Convergence Principle:

Cauchy Convergence Principle:

Pf.

  • Sufficiency:

Assume
Then

For any , we have

Hence, is a Cauchy sequence.

  • Necessity:
    First, prove that any Cauchy sequence is bounded. By the definition of a Cauchy sequence, Then Let Then

By the Bolzano-Weierstrass theorem, there exists a subsequence such that . Now, , by the Cauchy sequence definition,

Also,

For any , take sufficiently large such that . Then

Thus, .

Contraction Condition: If such that , then converges.

Pf. Show that is a Cauchy sequence. Assume without loss of generality that . Then:

3. Functional Limit

3.1 Definition and Properties

3.1.1 Definition of functional limit**

**Definition:** Suppose is defined in a certain deleted neighborhood of , i.e., , If , we have . Then is called the limit of at . Denoted as Note: is possible.

e.g. Prove

Pf.

3.1.2 Heine Theorem**

The necessary and sufficient condition for is that: for any sequence satisfying and , the corresponding sequence of functional values satisfies Pf. Necessity first. Assume , take any satisfying and . By the definition of functional limit, By the definition of sequential limit, Therefore , i.e., . Sufficiency. Proof by contradiction. Assume , i.e., , s.t. Take , s.t. and . Take , s.t. and . Obtain satisfying , but and
Theorem: The necessary and sufficient condition for the existence of is that for any sequence satisfying and , all converge.

3.2 Properties of Functional Limit

  • Uniqueness: If are both limits of at , then .
Pf. Take any and Then and . By the uniqueness of sequential limits, .
  • Squeeze Theorem: If , is satisfied. And , then .
Pf. Take any satisfying . Therefore . By Heine Theorem, . By the sequential Squeeze Theorem, . By Heine Theorem, .

  • Four Arithmetic Operations: Suppose .

Pf. Take any satisfying and . By Heine Theorem, . By the four arithmetic operations of sequential limits, By Heine Theorem,
  • Local Sign-Preserving Property: If and , then in a certain deleted neighborhood of , there always exists .
Pf. Proof by contradiction: Assume , s.t. Take , s.t. Take , s.t. Obtain a sequence satisfying and By Heine Theorem and the sign-preserving property of sequential limits, This contradicts .

Corollary:

  • If , then , s.t.
  • If , and , s.t. , , then .
  • If , then , s.t. is bounded in .

3.3 Single-Sided Limit

**Definition:** Let . If , it holds . Then is called the left-hand limit of at . Denoted as or .

Theorem. exists if and only if exist and are equal.

e.g. Heaviside Function

Functional Limit Definitions can be expanded. Consider a limit

“A” can be
“B” can be

e.g. .

3.4 Continuous Functions

3.4.1 Continuity

Definition. Suppose is defined in a neighborhood of , and , then is said to be continuous at .

expression: For , s.t. , holds.

Requirements:

  • Both left and right limits exist and are equal.
  • The function value exists and is equal to the limit.

(1) Continuity on an open interval.

If is continuous at every point in , then is said to be continuous on the open interval .

e.g. Prove is continuous on .

Take any , prove is continuous at . For , if , .

Let , then holds .
By definition, is continuous at . By the arbitrariness of , is continuous on .

(2) Continuity on a closed interval.

One-sided continuity: If , then is left-continuous.
If , then is right-continuous.

Continuity on a closed interval: If is continuous on , right-continuous at , and left-continuous at , then is said to be continuous on .

e.g. Prove is continuous on .

Take any , , we have:

then , we have thus is continuous on

Considering endpoints, , take , then , we have

Hence

Similarly

In summary, is continuous on .

If are continuous at the same point , then , , () are all continuous at .

Conclusion:

  1. Any polynomial is continuous on .
  2. Any rational function is continuous on its domain.

3.4.2 Discontinuity

(1) Types of Discontinuity Points

  1. Type I (Jump Discontinuity): Left and right limits exist but are not equal.

    • e.g.: , , etc.
    • Definition of jump magnitude:

  2. Type II: At least one of the left or right limits does not exist.

    • e.g.: ,

  3. Type III (Removable Discontinuity): The limit exists, but the function value does not exist or is not equal to the limit.

    • e.g.:

e.g.: Thomae’s Function (Riemann Function)

( are coprime)

Fact: . All points in are removable discontinuities, while all points in are continuity points.

Pf. . Take any . For , let . To prove that in the punctured neighborhood , there are only finitely many points satisfying the following condition ( is chosen arbitrarily):

For example, there will not be more points than those in the following table (taking the interval shown below):

Let the rational number satisfying condition (*) that is closest to be denoted as .
Let . For :

  1. If , then .
  2. If , the distance from to is closer than , i.e., .

By the definition of , does not satisfy condition (*). Thus , and .

(2) Discontinuities of Monotonic Functions

Theorem: Discontinuities of a monotonic function on must be Type I (jump) discontinuities.

Pf. Assume is monotonically increasing on . For , we want to show and exist.

Let . is an upper bound for . By the Least Upper Bound property, exists.

, since is no longer an upper bound for , there exists s.t. . Let . For , by monotonicity . By the definition of , .

Similarly, .

If , i.e., , we prove .

Proof by contradiction: Suppose . Since , then , . Similarly, , . Thus or .

① If , then . Contradiction.
② If , then . Contradiction.

In summary, the proposition is proven.

Theorem: A monotonic function has at most countably many discontinuities.

Pf. Without loss of generality, assume is monotonically increasing. Let be the set of all discontinuities of .

Since a monotonic function can only have Type I discontinuities, for , there exists an open interval . For , we must have , so we can find a series of disjoint open intervals for .

By the density of rational numbers, , there exists s.t. . Since rational numbers are countable, the set of disjoint open intervals is countable, and thus the set of discontinuities is countable.

(3) Continuity of Inverse Functions

If is strictly monotonic (increasing or decreasing), then has an inverse function , and is also strictly monotonic (increasing or decreasing).

Theorem: Suppose is strictly increasing and continuous on . Let . Then is a continuous function on .

Pf. First prove . Take any . To show s.t. . Let . By the Least Upper Bound property, let . For , there exists such that . By the definition of and monotonicity, . Since monotonic functions possess one-sided limits, . Similarly, it can be proven that . By the definition of continuous functions, . In summary, .

Next, prove continuity. Take any , and let . Take any , and let . Let . For , we have , which implies .

Finally, prove is right-continuous at . That is, , s.t. , . Take any , and let . By monotonicity, . Let . For , we have , hence . Similarly, it can be proven that is left-continuous at .

(4) Continuity of Composite Functions

Note: If and both exist, it does not necessarily follow that exists.

Theorem: If is continuous at , let , and is continuous at , then is continuous at .

Pf. , due to the continuity of at , s.t. , .
Furthermore, due to the continuity of , s.t. , .
In summary, , .
Thus, is continuous at .

3.5 Infinite Quantities

3.5.1 Infinitesimals and Their Orders

Definition (Infinitesimal): If , then is called an infinitesimal as .

Comparison: Suppose and are both infinitesimals as .

  • If , then is a higher-order infinitesimal of as .
    We denote this as .
  • If there exists , s.t. in a deleted neighborhood of , , then is bounded relative to as .
    We denote this as .

Theorem:

If there further exists , s.t. in a deleted neighborhood of , , then and are called infinitesimals of the same order.

  • If , then and are infinitesimals of the same order.
  • If , then and are equivalent infinitesimals, denoted as .

Order of Infinitesimals: If and are of the same order, we say is a -th order infinitesimal as .

  • is an infinitesimal.
  • is bounded.

As , if is an infinitesimal and and are infinitesimals of the same order, then is called a -th order infinitesimal as .

As , if is an infinitesimal, and and are infinitesimals of the same order, then is called a -th order infinitesimal as .

3.5.2 Comparison of Infinities

Definition (Infinity): If , then is called an infinity as .

Comparison: Suppose and are both infinities as .

  • If , then is a higher-order infinity of as .
  • If there exist , s.t. , , then is bounded relative to . We denote this as .
    Note: Higher-order infinity bounded.
  • If there exists , s.t. , , then and are called infinities of the same order as .
  • If , then and are equivalent infinities as , denoted as .

3.5.3 Equivalent Infinitesimals

(1) Common Equivalent Infinitesimals


  • Let . As , .


  • Let . As , .

(2) Applications of Equivalent Infinitesimals

Theorem: Suppose , , are defined in a deleted neighborhood of , and .

Pf.
.The rest is self-evident.

Theorem: If , then .

Pf.

Example:

3.6 Continuous Function on Close Interval

3.6.1 Boundness Theorem

If is continuous at , then , s.t. , .

Boundedness Theorem: If is continuous on the closed interval , then is bounded on .

Pf. (Using Nested Intervals Theorem): By contradiction. Suppose is unbounded on .
Let .
Bisect into and . At least one of these must have unbounded; denote it as .
Repeating this, we obtain a sequence where is unbounded on each .
By the Nested Intervals Theorem, , and .
Since , is continuous at . Thus, , s.t. is bounded on .
Since and , for any sufficiently large , .
This contradicts the fact that is unbounded on .

Pf. (Using Bolzano-Weierstrass Theorem): By contradiction. Suppose is unbounded on .
By the definition of unboundedness, , s.t. .
We obtain a sequence .
By the Bolzano-Weierstrass Theorem, , . Since is continuous, .
However, , which is a contradiction.

Pf. (Using Finite Covering Theorem):
By the local boundedness of continuous functions, , s.t. .
Let be an open covering of .
By the Finite Covering Theorem, .
Let .
Then , , s.t. .
Thus .
Therefore, is a bound for .

3.6.2 Extreme Value Theorem

Theorem (Extreme Value Theorem): If is continuous on , then there exist , s.t. for .

Pf. If exists, then is the supremum of .
Since continuous functions on closed intervals are bounded, by the Axiom of Completeness, exists.
Let . By the definition of supremum:
is not an upper bound of , so , s.t. .
We obtain a bounded sequence . By the Bolzano-Weierstrass Theorem, there exists a subsequence :
s.t. .
We have .
By the Squeeze Theorem and the continuity of , .
Thus , which means for .
The existence of can be proven similarly.

3.6.3 Zero Point Theorem

Theorem (Zero Point Theorem): If is continuous on and , then , s.t. .

Pf. (Using Nested Intervals Theorem): Assume without loss of generality and . Let .
If , the proof is complete.
If , let .
If , let .

Assume has been found such that and .
If , the proof is complete.
If , let .
If , let .

If the proof does not terminate in finite steps, we obtain a sequence of nested intervals satisfying .
By the Nested Intervals Theorem, , and .
By the continuity of , we have:

Since , the theorem is proven.

3.6.4 Fixed Points

(1) Fixed Point

Theorem: Suppose a continuous function on satisfies , then , s.t. (Fixed point of a function).

Pf.
.
Thus .
If or , the proof is complete.
If neither is 0, then . By the Zero Point Theorem, , s.t. , i.e., .

(2) Contraction Mapping Theorem

Definition (Contraction Mapping): , s.t. .

Contraction Mapping Theorem: If is a contraction mapping, then , s.t. .

Pf. 1:
is a continuous function. For , take , proof complete.
Let (to be determined).
, we have .
If , i.e., , then , which means .
By the Fixed Point Theorem, , s.t. .
Uniqueness: Assume are both fixed points.

Pf. 2:
By the Fixed Point Theorem, , s.t. .
Arbitrarily pick .
For , by the definition of contraction mapping: .

Then we have:

Since , for , we choose s.t. . Thus is a Cauchy sequence.
By the Cauchy Convergence Principle, , s.t. .
Since , by the continuity of , .
Thus is the fixed point.
If s.t. , then .

3.6.5 Intermediate Value Theorem

Theorem: If is continuous on , then it can take any value between and .

Pf.
By the Extreme Value Theorem, let such that and .
Without loss of generality, assume .
Pick any , and let .
Then and .
Applying the Zero Point Theorem to on , there exists such that .
That is, , and the theorem is proven.

Corollary: If is continuous on , , and .

3.6.6 Uniform Continuity

Continuity on Interval (Pointwise): .
depends not only on , but also on .

Definition (Uniform Continuity): Suppose is defined on interval . If , we have .

Intuitive Explanation: In general continuity, is a function of and . In uniform continuity, is independent of .

Example: is uniformly continuous on .
Contraction mapping is definitely uniformly continuous.

Theorem: is defined on . The necessary and sufficient condition for to be uniformly continuous on is:
.
Note: It is not required that or are convergent sequences.

Pf.
Necessity: Suppose is uniformly continuous, let .
, .
Since , .
Thus .

Sufficiency: By contradiction. Suppose is not uniformly continuous on .
, but .
Take .
We obtain , but .

Theorem (Cantor’s Theorem): If is continuous on , then is uniformly continuous on .

Pf.
Suppose is continuous on . For any , by the definition of continuity:
, such that , .
, we have:
.

forms an open covering of .

By the Finite Covering Theorem, .
Let .
Take any with . Since ,
.
Then .
Therefore . Q.E.D.

Theorem: Suppose is a finite open interval. If is continuous on , then the necessary and sufficient condition for to be uniformly continuous on is that the limits and exist.

Pf.

Sufficiency: .
Take a sequence such that .
.
By the uniform continuity of :
.
By Heine’s Theorem, .
Thus .
Therefore exists, and similarly, exists.

Necessity: By contradiction. Suppose is not uniformly continuous.
.
Take any , and let .
Take .
Take .
We obtain a sequence such that .
By Heine’s Theorem, does not converge to .
Due to the arbitrariness of , does not converge to .
Thus does not exist, which is a contradiction. Q.E.D.

4. Differentiation

4.1 Differentiation and Derivative

4.1.1 Differentiation

Suppose is defined in a neighborhood of . Consider the relationship between and :

  • is linear: is proportional to .
  • is non-linear but locally smooth at : When and are very small, .

Definition (Differentiability): Let . If there exists a function that depends only on and not on , such that as :

then is said to be differentiable at .

Basic Properties:

  • If is differentiable at , then as .
  • If , then as . is called the linear principal part of .

Definition (Differential): If is differentiable at , then as , is called the differential of the independent variable, denoted as .
is called the differential of the dependent variable, denoted as or .

Example: , find the differential.
Let the differential of the independent variable be .

.

Example: , differentiation at .
, which is a lower-order infinitesimal compared to .
is not differentiable at (if it were differentiable, would have to be a higher-order infinitesimal of ).

Note: Differentiable function must be continuous, but it’s not correct in the opposite direction.

4.1.2 Derivatives

If is differentiable at , then as .
Dividing by yields as .

Definition (Derivative): If and the limit exists, then is said to be derivable (or differentiable) at .
This limit is denoted as and is called the derivative of at .

Derivative Function: If is derivable at every point in an interval , then is derivable on .
The derivative at each point defines a new function for .

Theorem: is derivable at if and only if is differentiable at .

Pf.
Derivable Differentiable:
Suppose is derivable at . Then .
This implies .
Thus , which means .
Therefore , proving differentiability. Q.E.D.

3. Geometric Meaning of the Derivative

  • Secant line: The slope of the secant line is .
  • Tangent line: If exists, then has a tangent line at .
  • Equation of the tangent line: .

is derivable at is not equivalent to has a tangent line at (excluding the case where the tangent line is parallel to the -axis).

4.1.3 One-Sided Derivatives

Definition: If the limits or exist, then is said to be left-derivable or right-derivable at , respectively. The corresponding values are called the left derivative and right derivative.

Theorem: The necessary and sufficient condition for to be derivable at is that both $f’-(x_0)f’+(x_0)$ exist and are equal.
The derivation of this follows directly from the conditions for the existence of a limit.

e.g. Given , . is derivable at , and , .
Prove that .

Pf.
Since , we only need to prove .
Assume by contradiction that .
Then for the right derivative:

And for the left derivative:

$\implies f’+(\dfrac{1}{2}) \ne f’-(\dfrac{1}{2})fx = \dfrac{1}{2}$. Q.E.D.

4.2 Arithmetic Rules for Derivatives

4.2.1 Proofs of Common Derivatives

4.2.2 Arithmetic Rules for Derivatives

Theorem. Suppose and are both differentiable on a certain interval. Then:

Pf.
(1)

(2)

Let:

Then .

Generalizations.
(1)
(2)

Theorem. If is differentiable on and , then is differentiable on , and .

Pf.

Theorem. If are differentiable on and , then .

Pf.

4.2.3 Differentiation of Composite Functions

Theorem. Suppose is differentiable at and is differentiable at with . Then is differentiable at , and:

Proof. Let .
Since is continuous at , as , then .

Since , it follows that .
Thus, .

Formally: .

Theorem (Multiple Chain Rule): .

4.2.4 Differentiation of Inverse Functions

Theorem. Suppose is continuous and strictly monotonic on with . Let and . Then is differentiable on , and:

Proof. By the Inverse Function Continuity Theorem, is continuous on .

  • Strict Monotonicity.
  • Continuity.

Examples:
(1)

(2)

4.2.5 Differentiation of Determinants

Theorem.

Proof.
Let , where .

4.2.6 Special Differentiation Rules

(1) Logarithmic Differentiation

  • For forms like

  • For forms like

(2) Differentiation of Implicit Functions

Theorem (Invariance of the first-order differential form): .
Regardless of whether is an independent variable or an intermediate variable, .

Theorem. For an implicit function , differentiate both sides to get .
Example:

(3) Differentiation of Parametric Functions

Theorem.

Proof.


If , then (same order).

Alternative Method:

4.3 Higher-Order Derivatives and Differentials

4.3.1 Higher-Order Derivatives

Definition. If is differentiable at and is also differentiable at , then is said to be twice differentiable at . Similarly, -th order differentiability can be defined. It is denoted as:

Examples:

4.3.2 Operational Rules for Higher-Order Derivatives

  • Linearity:

  • Leibniz Formula:

Pf (by Mathematical Induction):
When , holds.
Assume the formula holds for .
Then for :

Example:

4.3.3 Higher-Order Derivatives of Composite Functions

Using the second-order derivative as an example:

Second-Order Derivative of Parametric Functions

4.3.4 Higher-Order Differentials

  • First-order differential:
  • Higher-order differentials differentiate the with respect to , rather than
  • is the independent variable, and is independent of , so .
  • Therefore:
  • Similarly, the -th order differential is:

Theorem: Higher-order differentials no longer possess “invariance of form.”

Pf. Let :

  • If is an independent variable, .

  • If is an intermediate variable, , :

  • If is an intermediate variable, then might not be .

4.4 Mean Value Theorems for Differentiation

4.4.1 Fermat’s Theorem

Definition (Extremum): Let . If there exists such that and for all , then is called a maximum point of , and is called a maximum value.

  • (The definition for minimum value is analogous)

Notes:

  • Extremum is a local concept.
  • A function may have infinitely many extrema within an interval.
  • Extrema are unrelated to continuity; they can also be dense.
  • Example:

Fermat’s Theorem: If is differentiable at and is an extremum point of , then .

Proof. Assume achieves a maximum at . Let such that .
And for all .

  • For , .
  • For , .

Since is differentiable at :

is a necessary but not sufficient condition.

4.4.2 Rolle’s Theorem

Theorem (Rolle’s Theorem): If is continuous on , differentiable on , and , then there exists at least one point such that .

Proof: By the Extreme Value Theorem, there exist such that for all .

  • If both and are at the endpoints, then . Thus, for all , .
  • If at least one of is not an endpoint:
    Suppose . Since is the minimum point of on , is a local minimum. By Fermat’s Theorem, .

Example: Legendre Polynomials: . Prove that has exactly roots in the interval .

Let

If , then .
has no zero points in .
By Rolle’s Theorem, has 1 zero point in .
…
has zero points in .

Generalized Rolle’s Theorem:
If is continuous and differentiable on , and (), then there exists such that .

Pf:

  • Case 1: .
    • If , the proof is complete.
    • If :
      Take such that . Without loss of generality, let .
      Let .
      Since :
      There exists such that for all , and for all , .
      This implies and .

Applying the Intermediate Value Theorem on , since :
There exists such that .
Similarly, there exists such that .
Applying Rolle’s Theorem on :
There exists such that .

  • Case 2: .
    Take such that (a finite value).
    Since :
    For all , there exists such that for all , .
    Take .

Applying the Intermediate Value Theorem on , there exists such that .
Similarly, there exists such that .
By Rolle’s Theorem, there exists such that .

4.4.3 Lagrange Mean Value Theorem

Theorem (Lagrange’s Theorem): Let be continuous on and differentiable on . Then there exists such that

Proof: Let ,
By Rolle’s Theorem, there exists such that ,

Precise Representation of :
, where

Differentiability:

If exists and is continuous at , then this formula is a precise relationship for

Theorem: If , then for all ,

Pf: Take . Without loss of generality, let
By Lagrange Theorem:

Theorem: Suppose is differentiable on an interval , is monotonically increasing , , , is strictly increasing

Pf:
Necessity of 1:
Sufficiency of 1: Take any ,
Apply Lagrange Theorem on
, s.t.

Proof of 2: Take any ,

On , by Lagrange Theorem:

Example: Prove

Example (Young’s Inequality): and , then
, , is concave
Let :

Theorem (Discontinuity of ): Suppose is differentiable everywhere on . Then can only have discontinuities of the second kind

Pf: Take any . We only need to prove that if and both exist, then is continuous at
By Lagrange Theorem:

Similarly,

Therefore, the existence of is continuous at .

4.4.4 Convexity of Functions

Definition: Let be defined on an interval . If for all , and for all , it holds that: . Then is called a convex function (downward convex) on . If the inequality is strict, it is called a strictly convex function.

Theorem: A convex function on an open interval is continuous.

Pf: Fix .
Let and .

Similarly,

Let , where .

Since ,

Similarly,

By the Squeeze Theorem, , hence is continuous at .

Theorem: Let be twice differentiable on interval . is convex on .

Pf (Necessity):
.
$\implies f’+(x_2) \geq f’+(x_1)f(x_1 + \Delta x) - f(x_1) \geq f(x_1 + \Delta x) - f(x_1)x_1 < x_2, \Delta x \in (0, x_2 - x_1)x_1 + \Delta x \in (x_1, x_2)$.

Case 2: .

Adding both equations:

Pf (Sufficiency):
Take .

where .
Since , then .
.

Theorem (Jensen’s Inequality): Let be a convex function on . For any and such that , then:

If , we obtain the Mean Value Inequality:

Inflection Points: If the convexity of changes on either side of , then is called an inflection point of the function’s graph.

Example:

Theorem: Let be continuous on , , be twice differentiable on , and has opposite signs on and . Then is an inflection point of .
The proof follows directly from the definition.

Theorem: Let be continuous on , , be twice differentiable on , and be an inflection point. Then .

Pf: ( is not necessarily a continuous function)
is an inflection point has opposite signs on either side of .
has opposite monotonicity on either side of .
is an extreme point of .
By Fermat’s Theorem, .

4.4.5 Cauchy Mean Value Theorem

Theorem: Suppose are continuous on , differentiable on , and . Then , s.t.

Pf:

By Rolle’s Theorem,

Example: is continuous on , differentiable on , and is bounded on .
Prove: is bounded on .
,

Only need to prove is bounded on .
,

Let , then

4.5 L’Hospital’s Rule

L’Hospital’s Rule: Suppose are continuous and differentiable on , and . If or (either one holds), and

then (The rule also holds for left-sided and two-sided limits).

Pf: Suppose .

. Extend the definition by letting .
By Cauchy’s Mean Value Theorem on ,

Taking the limit on both sides,


Fix any .

, by Cauchy’s Mean Value Theorem,

Note: L’Hospital’s Rule also applies to indeterminate forms

(In the above, are all convergent values)

The converse of L’Hospital’s Rule is not true.

Example: , does not exist.

4.6 Taylor Formula

4.6.1 Taylor’s Formula with Peano Remainder

Theorem: Suppose has an -th order derivative at . Then there exists a neighborhood of such that for any point in this neighborhood, the function value satisfies:

where as .

is an -th order polynomial, called the Taylor polynomial.
is called the Peano remainder.

Pf: Let .
It only needs to be proven that .
By L’Hospital’s Rule:
(-th order derivative exists at -th order derivative exists in )

Linear Form: .

4.6.2 Taylor’s Formula with Lagrange Remainder

Theorem: Suppose has -th order continuous derivatives on , and -th order derivatives on . Let be a fixed point. Then ,

where , and is between and .

Pf: Let .
To prove there exists between and , s.t.

The first order derivatives of are non-zero at points other than .
By Cauchy’s Mean Value Theorem ( times):

is between and .

Linear Form: .

4.6.3 Maclaurin Formula

Definition: The Taylor formula when is called the Maclaurin formula.

Examples:

4.6.4 Uniqueness of Taylor’s Formula

Theorem: Suppose is -th order differentiable at . Let be an -th order polynomial satisfying as . Then is the -th order Taylor polynomial of at .

Pf: Let be the -th order Taylor polynomial of at .
By the Peano remainder,

is an -th order polynomial

An -th order polynomial cannot have roots

Example: Taylor polynomial of at .

Note: If , then the Maclaurin formula can be used only if as .

Theorem: Suppose is -th order differentiable in a neighborhood of . Then its -th order Taylor polynomial is exactly the -th order Taylor polynomial of .

4.6.5 Applications

(1) Approximate Calculation

Example: Use the 10th order Taylor polynomial of to find and estimate the error.

(2) Finding Limits

Taylor’s formula for finding limits can be seen as an extension of equivalent infinitesimal substitution.

Example: Find

Example: Suppose is second-order continuously differentiable on . If exists and is bounded on , prove .

Only need to prove , if is large enough, then .
Suppose , satisfying .
Take . Since , there exists s.t. when , .
Take any , let . By Taylor’s formula for at :

(3) Proving Inequalities

Example: Suppose , prove when , .
Let , , .
.

Example: Suppose is second-order differentiable on . On , and .
Prove .
Taylor’s formula for at :

Subtracting the two equations:

(4) Finding Equations of Asymptotes for Curves

Horizontal Asymptote: is an asymptote .
Vertical Asymptote: is vertical .
Oblique: is an asymptote

Step 1: Find . If it exists and equals .
Step 2: Find . If it exists, .
If both exist, .

4.6.6 Taylor’s Formula with Cauchy Remainder

Theorem: Suppose has -th order continuous derivatives on , and let be a fixed point. Then , it holds that:

where the remainder is given by:

Pf:
When , the Newton-Leibniz formula holds.
Assume it holds for . Then for :

4.7 Applications

4.7.1 Local Extremum Problems

If is differentiable at , then is a necessary condition for to be a local extremum point (Fermat’s Theorem). Points where the derivative is are called stationary points (critical points).

Theorem: Suppose is defined in , continuous at , and differentiable in .

  • If and , then is a local minimum point (similarly for local maximum).
  • If has the same sign on both sides of , then is not a local extremum.

Theorem: Suppose is differentiable in , , and exists.

  • If , then is a local minimum point (convex downward) (similarly for local maximum).
  • If , the test is inconclusive.

Example: Find the extremum points of .

Local minimum points: .
Local maximum point: .

4.7.2 Global Extremum Problems

Consider continuous functions.

Example: Find the global extremum of the function on .
Extremum points: ; Boundary points: .

4.7.3 Function Graphing

The following steps are involved in graphing a function:

  • Domain and its points of discontinuity.
  • Monotonic intervals and local extremum points.
  • Concavity and inflection points.
  • Asymptotes.

5. Integrals

5.1 Indefinite Integrals

5.1.1 Concepts

Definition: If there exist two functions on a certain interval such that , then is called an antiderivative of on this interval. ( is still an antiderivative). The set of all antiderivatives of the function is called the indefinite integral. (The inverse operation of differentiation).

If , then:

and are inverse operations to each other:

Examples:

Note: Not all functions have an indefinite integral.

Example:

At , it is certainly not differentiable. , so does not have an indefinite integral on .

5.1.2 Properties of Indefinite Integrals

Theorem (Linearity): If and are antiderivatives of and respectively on a certain interval, then for any constants , is an antiderivative of . That is:

Example:

5.1.3 First Integration by Substitution (Change of Variables)

If , then:

Examples:

5.1.4 Second Integration by Substitution

Let . Then:

Example: Evaluate . Let .

5.1.5 Integration by Parts

Theorem:

Pf:

Examples:

Reduction Formula:

5.2 Integration of Rational Functions

5.2.1 Decomposition of Rational Functions

Definition: A function is defined as a rational function, where and are polynomials of degree and respectively.

When integrating rational functions, we consider the case where .
(If , then , where ).

Decomposition of Rational Functions:
The denominator can be factored into real roots and conjugate complex roots:

Theorem: The proper fraction () can be decomposed into a sum of the following rational functions:

Examples:

If you need the coefficient of a certain term, then multiple the denominator on both sides and make it zero.

5.2.2 Integration of Rational Functions

For quadratic denominators where :

Where:

And:

References:

[1] 陈纪修, 数学分析, 第三版. 北京: 高等教育出版社, 2019.
[2] В. А. Зорич, Математический анализ, части I, II, 7-е изд. Москва: МЦНМО, 2015. (中译本: 数学分析, 第一、二卷. 李植 译. 北京: 高等教育出版社, 2019.)

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On this page
Calculus
  1. 1. Set Theory and Functions
    1. 1.1 Generalized Distributive and De Morgan’s Laws
    2. 1.2 Fundamental Properties
  2. 2. Sequence
    1. 2.1 Continuity of Real Number
    2. 2.2 Sequence Limits and Infinitesimals
    3. 2.3 Arthematic Operations of Limit
    4. 2.4 Infinite Sequences
    5. 2.5 Convergence Criteria (Real Number Continuity Theorem)
    6. 2.7 Subsequence
    7. 2.8 Cauchy Convergence Principle
  3. 3. Functional Limit
    1. 3.1 Definition and Properties
    2. 3.2 Properties of Functional Limit
    3. 3.3 Single-Sided Limit
    4. 3.4 Continuous Functions
    5. 3.5 Infinite Quantities
    6. 3.6 Continuous Function on Close Interval
  4. 4. Differentiation
    1. 4.1 Differentiation and Derivative
    2. 4.2 Arithmetic Rules for Derivatives
    3. 4.3 Higher-Order Derivatives and Differentials
    4. 4.4 Mean Value Theorems for Differentiation
    5. 4.5 L’Hospital’s Rule
    6. 4.6 Taylor Formula
    7. 4.7 Applications
  5. 5. Integrals
    1. 5.1 Indefinite Integrals
    2. 5.2 Integration of Rational Functions
  6. References: