If $f$ has a derivative at $x=c$, then $f$ is continuous at $x=c$ #### Proof Given that $f^{\prime}(c)$ exists, we must show that $\lim _{x \rightarrow c} f(x)=f(c)$, or equivalently, that $\lim _{h \rightarrow 0} f(c+h)=f(c)$. If $h \neq 0$, then $ \begin{aligned} f(c+h) &=f(c)+(f(c+h)-f(c)) \\ &=f(c)+\frac{f(c+h)-f(c)}{h} \cdot h \end{aligned} $ Now take limits as $h \rightarrow 0$. By Theorem 1 of Section 2.2, $ \begin{aligned} \lim _{h \rightarrow 0} f(c+h) &=\lim _{h \rightarrow 0} f(c)+\lim _{h \rightarrow 0} \frac{f(c+h)-f(c)}{h} \cdot \lim _{h \rightarrow 0} h \\ &=f(c)+f^{\prime}(c) \cdot 0 \\ &=f(c)+0 \\ &=f(c) \end{aligned} $