Relation between continuity and differentiability 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:
Check here
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)
 A: 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.
A: No, you prove a function differentiable by evaluating the limit
$$\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$ and differentiability implies continuity (and not conversely, as the function $|x|$ at $x=0$ shows).

By contraposition, a discontinuous function is not differentiable, and proving the first may be simpler.
A: Differentiability implies continuity: since
$$
\lim_{x\to x_0} \frac{f(x) - f(x_0)}{x-x_0} 
$$
exists, $f(x) \to f(x_0)$ when $x\to x_0$. If not, then the limit does not exist, because
$$
\lim_{x\to x_0} x-x_0 = 0.
$$
So it’s not really necessary to check continuity first, but technically it’s often inevitable.
A: If a function is not continuous at a point $a$, it not differentiable at that point. Continuity does not guarantee differentiability, but continuity is necessary. If the function is not continuous, it's not differentiable, so checking for continuity can provide a preliminary check.
