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CONTINUITY & DIFFERENTIABILITY

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CONTINUITY & DIFFERENTIABILITY

Presented by

Muhammad Sarmad Hussain

Noreen Nasar

- Continuity
- Differentiability

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

exists &

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

- The function f
is continuous on a subset S of its domain,

if

it is continuous at every point of the subset.

- CLOSED FORM FUNCTIONS
- DOMAIN SPECIFIC FUNCTIONS
- NON-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

Examples of closed form functions are:

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.

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

Continuous but not Differentiable

Differentiable

Continuous but not 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.

- 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

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

- Consider the following examples

Domain of f(x) is THE bone of contention

- f(x) is differentiable throughout its domain

- f(x) is not differentiable throughout its domain

- Are all continuous functions differentiable ?

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

- Are all differentiable functions continuous ?

YES

- Mathematically provable and easy to understand.

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

So we can rewrite the equation as

exists &

THANKS