Continuous Function is not necessarily Differentiable

Theorem

Let $f$ be a real function defined on an interval $I$.

Let $x_0 \in I$ such that $f$ is continuous at $x_0$.

Then it is not necessarily the case that $f$ is differentiable at $x_0$.

Proof

Proof by Counterexample

Consider the real function $f: \R \to \R$ defined as:

$\forall x \in \R: \map f x = \size x$

where $\size x$ denotes the absolute value function.

From Absolute Value Function is Continuous, $f$ is continuous on $\R$.

From Derivative of Absolute Value Function, $f$ is not differentiable at $x = 0$.

Hence $f$ is an example of a real function which at $x_0 = 0$ is continuous at $x_0$ but not differentiable at $x_0$.

$\blacksquare$

Also see

• Differentiable Function is Continuous

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous function (continuous mapping)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuous function (continuous mapping, continuous map)