Confirmation of proof that differentiability implies continuity?

Question

I am trying to reason through the proof that differentiability implies continuity. I would appreciate some help reasoning through this one, if someone has a thorough explanation of the topic. I have tried to rewrite the definition of continuity to be equivalent to the definition of a derivative. My question is: does this work as a valid proof? If not, why? Thanks in advance!

Attempt at proof of concept.

Answer 1

As mentioned in the comments, if $f$ is diffentiable at $x$ then, $\lim_{t \to x} \frac{f(t) - f(x)}{t-x} = f'(x)$. So

$$\lim_{t\to x} f(t) - f(x) = \lim_{t \to x} \frac{f(t) - f(x)}{t-x}(t-x) = f'(x) \cdot 0$$

since both limits exist. Then given $\epsilon > 0$ from the definition of limit we can find a $\delta > 0$ such that $t \in (x - \delta, x + \delta)$ implies $|f(t) - f(x)| < \epsilon$. So $f$ is continuous at $x$.