Show that $f$ differentiable implies $f$ continous I have to show that, if a function $f:\mathbb R\to\mathbb R$ is differentiable, it is also continous.
$$\lim_{h\to0}\frac{f_{(x+h)}-f_x}{h}=f'_x\space\space\space\space\forall x\in \mathbb R$$
To show:
$$\lim_{h\to 0} f_{(x+h)}-f_x=0\space\space\space\space\forall x\in \mathbb R$$
I do not know hot to show that. My first thought was to set $h\to 0$:
$$\lim_{h\to 0} f_{(x+h)}-f_x=f_{(x+0)}-f_x=f_x-f_x=0?$$
But i think this is not a proper proof.
Can somebody help me?
Thank you

Let's consider a function $f: \mathbb{I} \rightarrow \mathbb{R}$. Also, let $f$ be differentiable at $c \in \mathbb{I}$. So, $f'(c)$ is finite. We know, that $\lim_{x \to c}{(x-c)}=0$. So, $$ \ \lim_{x \to c}{[f(x)-f(c)]}=lim_{x \to c}{\frac{f(x)-f(c)}{x-c}}(x-c)=f'(c)  \cdot 0=0$$ as, $f'(c)$ is finite. This shows that $ \lim_{x \to c}{f(x)}=f(c)$. Which means, $f$ is continuous at $c$.
