$f$ is differentiable in $\Bbb R \implies f$ is continuous in $\Bbb R$ Given a function $f:\Bbb R\to \Bbb R$, I need to show that if $f$ is differentiable then $f$ is continuous.
Here is my idea on how to show this: If $f$ is differentiable then $\forall x_0\in\Bbb R$, $f(x_0)$ is differentiable. Then this implies that $$\forall x_0\in\Bbb R, \lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)\iff \lim_{x\to x_0}[f(x)-f(x_0)]=\lim_{x\to x_0}[x-x_0]\frac{f(x)-f(x_0)}{x-x_0}.$$ Since $\lim_{x\to x_0}[x-x_0] = 0$, then from this we have, $$\lim_{x\to x_0}[f(x)-f(x_0)] = 0.$$ Moreover, since these limits exist independently, then we have $$\lim_{x\to x_0}f(x)=f(x_0)\implies f\text{ is continuous}$$  Does this seem correct or is there some error in the logic? Thanks.
 A: You seem like you have the right approach, but I am utterly confused by your symbols. You write an equality which means "$f$ is differentiable at $x_0$" then claim this is $\iff$ a statement about limits which is in fact necessarily true purely from the cancellation of $[x - x_0]$ from the top and bottom of that fraction. One could argue about what it even means if the limit does not exist, but certainly if $f$ is merely continuous it does, so the RHS of your $\iff$ is more often true than the LHS.
So as stated it's not correct. But I suspect you have the correct proof in your brain, and are just writing down the wrong things. 
A: @Ben Millwood
I am posting this as an answer because it would be too much for a comment.  Anyways, what I was saying is this: $$\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)$$*is logically equivalent to* $$\lim_{x\to x_0}[f(x)-f(x_0)]=\lim_{x\to x_0}[x-x_0]\lim_{x\to x_0}f'(x_0), \text{where}\lim_{x-x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0), \text{etc...}$$  I was just cutting out some additional steps
