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CONTINUITY & DIFFERENTIABILITY PowerPoint PPT Presentation

CONTINUITY & DIFFERENTIABILITY. Presented by Muhammad Sarmad Hussain Noreen Nasar. Overview. Continuity Differentiability. Continuous at a Point. A function f is continuous at point a in its domain , if. exists &. Domain.

CONTINUITY & DIFFERENTIABILITY

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CONTINUITY & DIFFERENTIABILITY

Presented by

Noreen Nasar

Overview

• Continuity

• Differentiability

Continuous at a Point

• A function f is continuous at point a in its domain, if

exists &

Domain

• If the point a is not in the domain of f, we do not talk about whether or not f is continuous at a.

Continuous on Subset of Domain

• The function f

is continuous on a subset S of its domain,

if

it is continuous at every point of the subset.

EXAMPLES

• CLOSED FORM FUNCTIONS

• DOMAIN SPECIFIC FUNCTIONS

• NON-CLOSED FORM FUNCTIONS

Closed Form Functions

All functions are continuous on their (whole) domain. A closed-form function is any function that can be obtained by combining

• constants,

• powers of x,

• exponential functions,

• logarithms, and trigonometric functions

Closed Form Functions

Examples of closed form functions are:

Domain Specified Functions

The function f(x) = 1/x is continuous at every point of its domain.

Note that 0 is not a point of the domain

of f, so we don't discuss what it might

mean to be continous or discontinuous

there.

Non – Closed Form Functions

The function

f(x)   = -1if x ≤ 2

if x > 2

is not a closed-form function

Because we need two algebraic formulas

to specify it. Moreover, it is not continuous

at x = 2, since limx 2f(x) does not exist

Discontinuous

Discontinuos

Continuous

Continuous

Undefined

Continuous

Discontinuous

Undefined

Discontinuos

Continuous

Undefined

Graphical Representation

Continuous but not Differentiable

Graphical Representation

Differentiable

Continuous but not Differentiable

Differentiable

• f is differentiable at point “a” if f’(a) exists

• f is differentiable on the subset of its domain if it is differentiable at each point of the subset.

Undifferentiable

• f is not differentiable when

• The limit does not exist, i.e.

does not exist.

This situation, when represented in graphical form leads to a cusp in the graph

Undifferentiable

• Limit goes to infinity, e.g. in the following case

Isolated Non – Differentiable Points

• Consider the following examples

Difference of Opinion

Domain of f(x) is THE bone of contention

• f(x) is differentiable throughout its domain

• f(x) is not differentiable throughout its domain

The Question

• Are all continuous functions differentiable ?

Another Question

• If f is not continuous at point a, then is it differentiable at that point ?

One More

• Are all differentiable functions continuous ?

YES

• Mathematically provable and easy to understand.

Proof of Continuity

• Suppose f(x) is differentiable at x=a

Contd…..

So we can rewrite the equation as

exists &

THANKS