Differentiable Function is Continuous

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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 $x_0$.


Corollary

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


Proof

By hypothesis, $\map {f'} {x_0}$ exists.

We have:

\(\displaystyle \map f x - \map f {x_0}\) \(=\) \(\displaystyle \frac {\map f x - \map f {x_0} } {x - x_0} \cdot \paren {x - x_0}\)
\(\displaystyle \) \(\to\) \(\displaystyle \map {f'} {x_0} \cdot 0\) as $x \to x_0$


Thus:

$\map f x \to \map f {x_0}$ as $x \to x_0$

or in other words:

$\displaystyle \lim_{x \mathop \to x_0} \map f x = \map f {x_0}$


The result follows by definition of continuous.

$\blacksquare$


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