# Differentiability Implies Continuity

A theorem in Calculus states that if $f$ is differentiable at $x = c$, then $f$ is continuous at $x = c$. With this theorem in mind it can be proven that $f$ is continuous at $x = c$ by showing that $f(x)$ approaches $f(c)$ as $x \rightarrow c$. To do this, we can use the differentiability of $f$ at $x = c$ and consider the following limit.

$\displaystyle\lim_{x\to\ c} [f(x) - f(c)] = \displaystyle\lim_{x\to\ c} \bigg[(x - c)\bigg( \frac{f(x) - f(c)}{x - c} \bigg) \bigg]$

$= \bigg[\displaystyle\lim_{x\to\ c} (x - c) \bigg] \bigg[\displaystyle\lim_{x\to\ c} \frac{f(x) - f(c)}{x - c}\bigg]$

$= (0)[f\prime(c)]$

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