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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I'm not sure what the question is. Starting from the assumption that f is differentiable at f(x0) and proving that it's continuous at f(x0) does prove that differentiability implies continuity.
That's what implication means.
Haha yeah I thought about it some more and see it now. It was a long day when I asked the question. Continuous means that the (derivative of f(x)) - f(x0) = 0 and it's legimate to subtrace that f(x0) from any differentiable function. But doing that we see when end up the fact that it equals zero so we know any differentiable function is continuous at x0