# Continuous not implies differentiable

This article gives the statement and possibly, proof, of a non-implication relation between two function properties. That is, it states that every function satisfying the first function property (i.e., continuous function) neednotsatisfy the second function property (i.e., differentiable function)

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## Contents

## Statement

It is possible to have a function that is a continuous function everywhere but is *not* a differentiable function everywhere, i.e., it has points where it is not differentiable.

Further, it is possible to construct an example where the function does not have a well-defined left-hand derivative or right-hand derivative.

## Proof

### Example where one-sided derivatives exist and are not equal

A piecewise definition by interval can be used to construct examples (see differentiation rule for piecewise definition by interval). The simplest example is the absolute value function:

The function is continuous at 0. The left-hand derivative at 0 is -1 and the right-hand derivative is 1. These are not equal, so the function is not differentiable at 0.

Intuitively, the function is continuous at the point 0 where its definition changes, but it is not differentiable because it turns sharply (from moving down to moving up). There is no well-defined tangent line at (0,0) where it turns.

### Example where no one-sided derivatives exist

Consider the function:

The expression for the function is continuous and differentiable for . The expression does not extend to , so we need to calculate the limit at 0 manually to determine if it is continuous and differentiable.

### Proof of continuity at 0

We have a bound on absolute value:

Thus:

Both the leftmost and rightmost expressions approach 0 as , so by the sandwich theorem, we get:

### Proof of non-differentiability at 0

The derivative at zero is the limit:

This works out to:

The sine of reciprocal function does not have a limit as , so this expression is not defined. In fact, even restricting to a one-sided limit gives an undefined expression (for more on this, see the explanation at intermediate value property not implies continuous).

## Related facts

- Intermediate value property not implies continuous: This uses the sine of reciprocal function directly, i.e., just .
- Derivative of differentiable function need not be continuous: This uses .