In quantum mechanics we are told that

1. the wavefunctions live in Hilbert space.
2. the wavefunctions are continuous.

It recently came to my notice that in Mathematics, there is a theorem which says that the $\textit{continuous functions}$, for example $C[0,1]$ do not form a Hilbert space (see [this](https://math.stackexchange.com/q/112168/)). Now I wonder whether this is in contradiction with 1 and 2 or not?