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CONTINUITY & DIFFERENTIABILITYPowerPoint Presentation

CONTINUITY & DIFFERENTIABILITY

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## PowerPoint Slideshow about ' CONTINUITY & DIFFERENTIABILITY' - frances-gonzalez

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Presentation Transcript

Overview

- Continuity
- Differentiability

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) = -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

Graphical Representation

Continuous

Graphical Representation

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

Continuity

exists &

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