# Differentiability and Continuity

**If a function is differentiable at point x = a, then the function is continuous at x = a.**

The reciprocal may not be true, that is to say, there are functions that are continuous at a point which, however, may not be differentiable.

#### Examples

Study the continuity and differentiability of the following functions:

First, study the continuity at x = 0.

The function is not continuous, therefore it is not differentiable.

First, study the continuity at x = 0.

This function is continuous, so the differentiability can be studied.

It is not differentiable at x = 0.

**f(x) = x ^{2} at x = 0.**

The function is continuous at x = 0, so the differentiability can be studied.

At x = 0, the function is continuous and differentiable.

Subject

Site