As I understand, a smooth function is continuously differentiable.
But if I have a function which is continuous AND differentiable, I cannot automatically say that it is smooth. For it has to be so for all its differentials.
So I wonder, what function would be continuous and differentiable, but not continuously differentiable?
I cannot find the answer myself, as I do not clearly understand the difference between continuous AND differentiable, and continuously differentiable....
Context: I ask this because of an arc length contest. The function has to be continuous and differentiable on [0,1]. But does this automatically mean that I may always use the formula for an arc length, which has the condition that the function is smooth... (or, in another book, that it has a continuous derivative)?