Continuity differentiability
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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

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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)   = -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

Discontinuous


Discontinuous1

Discontinuous

Discontinuous

Discontinuos


Graphical representation

Graphical Representation


Graphical representation1

Graphical Representation

Continuous


Graphical representation2

Graphical Representation

Continuous

Undefined


Graphical representation3

Graphical Representation

Continuous

Discontinuous

Undefined


Graphical representation4

Graphical Representation


Graphical representation5

Graphical Representation

Discontinuos

Continuous

Undefined


Graphical representation6

Graphical Representation


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


Undifferentiable1

Undifferentiable


Undifferentiable2

Undifferentiable

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


Isolated non differentiable points

Isolated Non – Differentiable Points

  • Consider the following examples


Isolated non differentiable points1

Isolated Non – Differentiable Points


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 &


Confusions questions

Confusions Questions


Continuity differentiability

THANKS


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