TITLE: What is the difference between continuous derivative and derivative?
QUESTION [20 upvotes]: What is the difference between continuous derivative and derivative?
According to my teachers solution to the assignment,it seems there exits difference between continuous derivative and derivative. However, aunt Google does not tell me what I want.
Edit:
Here is a example.
$$f(x)
= \begin{cases}
\frac{1-cos2x}{x} & \text{otherwise}  \\
k & \text{if x=0} 
\end{cases}$$
Does k is continuous but not continuous derivative at $0$?
Thanks:)

REPLY [5 votes]: A function needs to be continuous in order to be differentiable. However the derivative is just another function that might or might not itself be continuous, ergo differentiable.