I have noticed that authors of introductory analysis textbooks always show that a function f is continuous at a point before trying to prove that it is differentiable there. Ahlfors, for instance, does this in "Complex Analysis: An Introduction to The Theory of Analytic Functions of One Complex Variable" when he proves the extended Cauchy's integral formula by induction:
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Bartle does the same when proving the second form of the Fundamental theorem of calculus in "Introduction to Analysis". Is it always necessary to prove continuity at a point before showing that a function is differentiable there? Why? (Please provide a technical explanation)
Best Answer
Being differentiable in a point is stronger than being continuous in that point (this means that being differentiable in a point implies being continuous in that point) Hence it is just make sense to start from the simpler attribute. Moreover, (maybe someone will correct me here), as being differentiable means (at least in the common cases) that the following exists: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ then, the logical requirement is that when $h$ goes to $0$ you want $f(x+h)$ to become close to $f(x)$, and that is continuity.