Differentiable Function is Continuous

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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 $\xi$.

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

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

We have:

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle f \left({x}\right) - f \left({\xi}\right)$	$=$	$\displaystyle \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi} \cdot \left({x - \xi}\right)$	$\displaystyle $	$\displaystyle $	$\displaystyle $
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\to$	$\displaystyle f' \left({\xi}\right) \cdot 0$	$\displaystyle $	$\displaystyle $	$\displaystyle $	as $x \to \xi$

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.

$\blacksquare$

The corollary is the contrapositive of the main theorem.

By the Rule of Transposition, the corollary holds.

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle f \left({x}\right) - f \left({\xi}\right)\)	\(=\)	\(\displaystyle \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi} \cdot \left({x - \xi}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\to\)	\(\displaystyle f' \left({\xi}\right) \cdot 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	as $x \to \xi$