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, $f' \left({x_0}\right)$ exists.

We have:

Thus:
 * $f \left({x}\right) \to f \left({x_0}\right)$ as $x \to x_0$

or in other words:
 * $\displaystyle \lim_{x \to x_0} \ f \left({x}\right) = f \left({x_0}\right)$

The result follows by definition of continuous.