I just read a proof of the fact that the space of functions which are differentiable at at least one $x \in [0,1]$ is nowhere dense in $C([0,1])$. However, I've also read proofs that say that the polynomials are dense in $C([0,1])$ (Weierstrass approx. theorem) and they are most certainly differentiable at at least one $x \in [0,1]$. In fact, the proof I just read even goes so far to claim that piecewise affine functions (which are almost everywhere differentiable) are dense in $C([0,1])$. I think I must be missing something here.
On one hand, it seems that polynomials and piecewise affine functions should be subsets of the space of functions differentiable at at least one $x \in [0,1]$, from which I should be able to conclude that they are also nowhere dense. So, is it possible for a subset of a nowhere dense set in $C[0,1]$ to be dense in $C([0,1])$?