Suppose we have a continuous function f:R→R. Suppose also that for a certain point c, lim(x→c)f′(x) exists. Must f′(c) exist as well, and be equal to this limit?
This isn't quite the same as asking if derivatives are always continuous. The well-known function f(x)=x^2*sin(1/x) is continuous and differentiable everywhere, but its derivative has no limit at x=0. I'm wondering if the derivative of a continuous function can have a discontinuity where its limit does exist.
I know this is a duplicate.. but the original question has a wrong answer. It assumed that f is differentiable (except at x=a) at an open interval which containing a .