Is every differentiable function continuous?

Yes, if a function is differentiable at a point then it is necessary continuous at that point.

Proof : Let a function $f(x)$ be differentiable at $x=c$. Then,

$\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}$ exists finitely.

Let $\lim _{x \rightarrow c} \frac{f(x)-f(c)}{x-c}=f^{\prime}(c)$

In order to prove that $f(x)$ is continous at $x=c$, it is sufficient to show that $\lim f(x)=f(c)$

$\lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left\{\left(\frac{f(x)-f(c)}{x-c}\right)(x-c)+f(c)\right\}$

$\Rightarrow \lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left[\left\{\frac{f(x)-f(c)}{x-c}\right\}(x-c)\right]+f(c)$

$\Rightarrow \lim _{x \rightarrow c} f(x)=\lim _{x \rightarrow c}\left\{\frac{f(x)-f(c)}{x-c}\right\} \cdot \lim _{x \rightarrow c}(x-c)+f(c)$

$\Rightarrow \lim _{x \rightarrow c} f(x)=f^{\prime}(c) \times 0+f(c)$

$\Rightarrow \lim _{x \rightarrow c} f(x)=f(c)$

Hence, $f(x)$ is continuous at $x=c$.