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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.

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Continuity differentiability

CONTINUITY & DIFFERENTIABILITY

Presented by

Muhammad Sarmad Hussain

Noreen Nasar


Overview
Overview

  • Continuity

  • Differentiability


Continuous at a point
Continuous at a Point

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

exists &


Domain
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
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
EXAMPLES

  • CLOSED FORM FUNCTIONS

  • DOMAIN SPECIFIC FUNCTIONS

  • NON-CLOSED FORM FUNCTIONS


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 functions1
Closed Form Functions

Examples of closed form functions are:


Domain specified functions
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
Non – Closed Form Functions

The function

f(x)   = -1 if 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



Discontinuous1
Discontinuous

Discontinuous

Discontinuos




Graphical representation2
Graphical Representation

Continuous

Undefined


Graphical representation3
Graphical Representation

Continuous

Discontinuous

Undefined



Graphical representation5
Graphical Representation

Discontinuos

Continuous

Undefined



Graphical representation7
Graphical Representation

Continuous but not Differentiable


Graphical representation8
Graphical Representation

Differentiable

Continuous but not Differentiable


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
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



Undifferentiable2
Undifferentiable

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


Isolated non differentiable points
Isolated Non – Differentiable Points

  • Consider the following examples



Difference of opinion
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
The Question

  • Are all continuous functions differentiable ?


Another question
Another Question

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


One more
One More

  • Are all differentiable functions continuous ?

YES

  • Mathematically provable and easy to understand.


Proof of continuity
Proof of Continuity

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


Contd
Contd…..

So we can rewrite the equation as


Continuity
Continuity

exists &