I don't get the proof of Differentiable implies continuous. We prove that lim f(x) as x tends to x0 - f(x0) = 0 (which is the definition of continuous) by manipulating it to be f'(x) * 0. I don't see how this completes a proof. This just proves that f is continuous. In fact, in the accompanying notes it says that this shows "it’s true that lim f(x) (as x tends to x0) −f(x0) = 0, and it’s true that diﬀerentiable functions are continuous." I don't see where differentiable implies continuous in all of this. What am I missing?

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