## Advanced Calculus (I)

WEN-CHINGLIEN

Department of Mathematics National Cheng Kung University

## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈**R**
if and only if f is defined on some open interval I

containing a and

(1) f^{0}(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f^{0}(a) is called the derivative of f at a.

**Notation:**

Dxf = df
dx =f^{(1)}=f^{0}

## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈**R**
if and only if f is defined on some open interval I

containing a and

(1) f^{0}(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f^{0}(a) is called the derivative of f at a.

**Notation:**

Dxf = df
dx =f^{(1)}=f^{0}

## 4.1 The Derivative

Definition

A real function f is said to be differentiable at a point a ∈**R**
if and only if f is defined on some open interval I

containing a and

(1) f^{0}(a) = lim

h→0

f (a + h) − f (a) h

exists. In this case f^{0}(a) is called the derivative of f at a.

**Notation:**

Dxf = df
dx =f^{(1)}=f^{0}

Theorem

A real function f is differentiable at some point a ∈**R if and**
only if there exists an open interval I and a function

F : I →**R such that a ∈ I, f is defined on I, F is continuous**
at a, and

(3) f (x ) = F (x )(x − a) + f (a)
holds for all x ∈ I, in which case F (a) = f^{0}(a).

Theorem

A real function f is differentiable at some point a ∈**R if and**
only if there exists an open interval I and a function

F : I →**R such that a ∈ I, f is defined on I, F is continuous**
at a, and

(3) f (x ) = F (x )(x − a) + f (a)
holds for all x ∈ I, in which case F (a) = f^{0}(a).

Theorem

Let f :**R → R. Then f is differentiable at a if and only if**
there is a function T of the form T (x ) := mx such that

(4) lim

h→0

|f (a + h) − f (a) − T (h)|

|h| =0.

Theorem

Let f :**R → R. Then f is differentiable at a if and only if**
there is a function T of the form T (x ) := mx such that

(4) lim

h→0

|f (a + h) − f (a) − T (h)|

|h| =0.

Theorem

If f is differentiable at a, then f is continuous at a.

Theorem

If f is differentiable at a, then f is continuous at a.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if
f^{0}I exists and is continuous on I.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if
f^{0}I exists and is continuous on I.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

f is said to be continuously differentiable on I if and only if
f^{0}I exists and is continuous on I.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

^{0}I exists and is continuous on I.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

^{0}I exists and is continuous on I.

Definition

Let I be a nondegenerate interval.

(i)

A function f : I →**R is said to be differentiable on I if and**
only if

f^{0}I(a) = lim

x →a x ∈I

f (x ) − f (a) x − a exists and is finite for every a ∈ I.

(ii)

^{0}I exists and is continuous on I.

**Example:**

The function

f (x ) = (

x^{2}sin(1

x) x 6= 0

0 x = 0

is differentiable on**R but not continuously differentiable on**
any interival that contains the origin.

**Example:**

The function

f (x ) = (

x^{2}sin(1

x) x 6= 0

0 x = 0

is differentiable on**R but not continuously differentiable on**
any interival that contains the origin.

**Remark:**

f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].

**Remark:**

f (x ) = |x | is differentiable on [0, 1] and on [−1, 0] but not on [−1, 1].

**Proof:**

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

Since f (x ) = x when x > 0 and = −x when x < 0,it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5,f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0.However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

Since f (x ) = x when x > 0 and = −x when x < 0, it is clear that f is differentiable on [−1, 0) ∪ (0, 1] (with

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5,f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0.However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

f^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore,f is differentiable on [0, 1] and on [−1, 0]. 2

**Proof:**

^{0}(x ) = 1 for x > 0 and f^{0}(x ) = −1 for x < 0). By example
4.5, f is not differentiable at x = 0. However,

f^{0}[0,1](0) = lim

h→0+

|h|

h =1 and f^{0}[−1,0](0) = lim

h→0−

|h|

h =1 Therefore, f is differentiable on [0, 1] and on [−1, 0]. 2