Let $X\subseteq\mathbb R^n$ be an affine variety (that is, a set given by polynomial equations). One may suppose $X$ singular (the case of non-singular $X$ is obvious).
Is it true that any continuous map $[0,1]^m\to X$ can be $C^0$-approximated by a smooth one?
A map $[0,1]^m\to X$ is called smooth if its composition with the inclusion $$ [0,1]^m\to X\to\mathbb R^n $$ is smooth (that is, $C^\infty$).