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 $$\lim_{x \to \xi} f \left({x}\right) = f \left({\xi}\right)$$.

The result follows by definition of continuous.