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 $\xi$.

Corollary
If $f$ is not continuous at $\xi$, $f$ is not differentiable at $\xi$.

Proof
By hypothesis, $f' \left({\xi}\right)$ exists.

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.

Proof of Corollary
The corollary is the contrapositive of the main theorem.

By the Rule of Transposition, the corollary holds.