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:

Piecewise Function

First, study the continuity at x = 0.

Continuity

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

Piecewise Function


Function

First, study the continuity at x = 0.

Continuity at x = 0

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

Differentiability and Continuity

Differentiability and Continuity

It is not differentiable at x = 0.

Differentiability and Continuity


f(x) = x2 at x = 0.

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

Differentiability and Continuity

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

Function