Some theorems can reach a conclusion when the derivative of some real-valued function of a real variable is continuous, and some other theorems can reach a conclusion if the derivative merely exists. So, I guess that existence and continuity are not equivalent statements. But thinking about the definition of the derivative, my intuition is telling me that existence implies continuity. Or, equivalently, discontinuity implies nonexistence. My intuition visualizes a discontinuous derivative as a sharp corner in a curve. But a sharp corner looks to me like an undefined derivative at the corner. Am I wrong? I think I am wrong, because the literature makes a distinction between existence and continuity, but I don't understand how I can be wrong. I am looking for an example of a function that is differentiable at every point in some interval but with the derivative discontinuous at some point in that interval. Can you give an example?
Hi Larry. Consider a function example for this case, f(x) being defined in two parts:
x^2 sin(1/x) for x != 0,
0 for x = 0.
For the domain, take an interval including zero.
The function f is differentiable at every point in the domain; to see this particularly in the pathological zero point, apply the limit definition of the derivative.
Moreover, the limit of the derivative of the function, f', when the independent variable approaches zero, does not exist (it oscillates). Then, f' is not continuous there.
Thank you Sebastian. That is exactly the kind of answer that I was looking for.