Differentiable Function is Continuous

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

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

Then $f$ is continuous at $\xi$.

Proof
We have:

Thus $f \left({x}\right) \to f \left({\xi}\right)$ as $x \to \xi$ or $\displaystyle \lim_{x \to \xi} f \left({x}\right) = f \left({\xi}\right)$.

The result follows by definition of continuous.