Which functions are $C^k$, but not $C^{k+1}$ on $ \mathbb{R} $ The only functions I can think of that fulfill this property are the polynomials of degree k. 
However this does not necessarily imply, that this are the only functions that are in such a space. So I am curious, if there exist some further characterization of this set. 
So I would be very happy about any constructive comment, answer or recommendation for further reading. As always thanks in advance.   

Most (in the sense of Baire category) continuous functions are nowhere differentiable. See Most Continuous Functions are Nowhere Differentiable.
Given any continuous nowhere differentiable function ($C^0$ and not $C^1$), any primitive will be $C^1$ but not $C^2$...
