Okay, so in class today we went over proving continuity by the formal definition of limits, that is
$$\forall \epsilon \gt 0, \exists \delta \gt 0 \mid \lvert x-a \rvert \lt \delta \implies \lvert f(x)-f(a)\rvert\lt\epsilon$$
So, that's all fine. But my professor gave an example where $f(x)=\ln(x)$ and he tried to show continuity at $x=2$.
Now, he proved it using the MVT:
$$\lvert\ln(x)-\ln(2)\rvert = \lvert f'(c) \left(x-2\right)\rvert$$
Am I crazy to think this cannot be used? For the MVT, you are assuming continuity, but we are trying to prove continuity!
Also, if MVT cannot be used, can someone show how to prove this?