# How do you prove a function is differentiable if it is continuous?

Jun 22, 2019

## How do you prove a function is differentiable if it is continuous?

Complete step-by-step solution: Let f(x) is function which is differentiable at point x=c, so according to differentiability definition f′(c)=limx→cf(x)−f(c)x−c Exists. Also when a function is continuous at point x=c then limx→af(x)=f(c) exists.

## What is continuously differentiable?

A function is said to be continuously differentiable if the derivative exists and is itself a continuous function. Although the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.

How do you prove a function is differentiable?

A function is said to be differentiable if the derivative of the function exists at all points in its domain. Particularly, if a function f(x) is differentiable at x = a, then f′(a) exists in the domain.

What’s the difference between continuous and differentiable?

The difference between the continuous and differentiable function is that the continuous function is a function, in which the curve obtained is a single unbroken curve. It means that the curve is not discontinuous. Whereas, the function is said to be differentiable if the function has a derivative.

### Can a function be continuous and not differentiable?

We see that if a function is differentiable at a point, then it must be continuous at that point. If is not continuous at , then is not differentiable at . Thus from the theorem above, we see that all differentiable functions on are continuous on .

### How do you prove something is not differentiable?

In each case, the easiest thing will be to consider the one sided limits, as h→0+ and as h→0−; if you can show that the one-sided limits are different from each other or at least one does not exist (including the case that they equal ∞ or −∞), (each of the two limits separately, of course), then this will prove the …

Can a graph be continuous but not differentiable?

What is continuous but not differentiable?

In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.

#### Are all continuous functions differentiable?

We have the statement which is given to us in the question that: Every continuous function is differentiable. Therefore, the limits do not exist and thus the function is not differentiable. But we see that f(x)=|x| is continuous because limx→cf(x)=limx→c|x|=f(c) exists for all the possible values of c.

#### What does it mean to say a function is continuous but nowhere differentiable?

It is a continuous, but nowhere differentiable function, defined as an infinite series: f(x) = SUMn=0 to infinity Bn cos (An * Pi * x) where A and B can be any numbers such that B is between 0 and 1, and A * B is bigger than 1 + (3 * Pi/2). For instance, A=12, B=1/2 will work.

What types of functions are not differentiable?

The four types of functions that are not differentiable are: 1) Corners 2) Cusps 3) Vertical tangents 4) Any discontinuities Page 3 Give me a function is that is continuous at a point but not differentiable at the point. A graph with a corner would do.

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Which is an example of a continuous differentiable function?

No, they are not equivalent. A function is said to be differentiable at a point if the limit which defines the derivate exists at that point. However, the function you get as an expression for the derivative itself may not be continuous at that point. A good example of such a function is x2(sin(1 / x2), which has a finite derivative at x = 0,…