Differentiable Function is Continuous

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

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

Then $f$ is continuous at $x_0$.

Proof
By hypothesis, $\map {f'} {x_0}$ exists.

We have:

Thus:
 * $\map f x \to \map f {x_0}$ as $x \to x_0$

or in other words:
 * $\ds \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$

The result follows by definition of continuous.

Also see

 * Complex-Differentiable Function is Continuous