TITLE: Does there exist a continuous and differentiable function which isn't smooth?
QUESTION [8 upvotes]: 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)?

REPLY [9 votes]: We say that $f$ is a function of class $C^k$ if it has $k$ continuous derivatives. You want to find a differentiable function which is not $C^1$, that is, a differentiable function with discontinuous derivative. Given $k \in \Bbb Z_{> 0}$, the function $$f(x) = \begin{cases} x^k\sin(1/x), & \text{if } x \neq 0 \\ 0, & \text{if }x=0\end{cases}$$ is $C^{k-1}$ but not $C^k$.