Differentiable Function is Continuous

This article was Featured Proof between 3rd October 2012 and 20 October 2012.

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 $x_0$.

If $f$ is not continuous at $x_0$, $f$ is not differentiable at $x_0$.

We have by hypothesis that $\map {f'} {x_0}$ exists.

Let $x, x_0 \in I$ such that $x \ne x_0$. Then:

$\ds \map f x - \map f {x_0}$

$=$

$\ds \frac {\map f x - \map f {x_0} } {x - x_0} \cdot \paren {x - x_0}$

$\ds $

$\to$

$\ds \map {f'} {x_0} \cdot 0$

as $x \to x_0$

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.

$\blacksquare$