# Derivative of differentiable function need not be continuous

From Calculus

## Statement

It is possible to have a function defined for real numbers such that is a differentiable function everywhere on its domain but the derivative is not a continuous function.

Equivalently, a differentiable function on the real numbers need not be a continuously differentiable function.

## Proof

### Example with an isolated discontinuity

Consider the function:

Then, we have:

In particular, we note that but does not exist. Thus, is not a continuous function at 0.

For details, see square times sine of reciprocal function#First derivative.

The video below covers this example.