Given $A,B$ two normed finite dimensional vector spaces and $f: U\subset A \to B$ a continuous map (where $U$ is open in $A$); show that, $f$ is differentiable ($C^{\infty}$) if and only if $f^{*}(C^\infty(B))\subseteq C^{\infty}(U)$. Here, $f^*(g)= g\circ f$.
It's easy to see that if $f$ is $C^\infty$ then, $f^{*}(C^\infty(B))\subseteq C^{\infty}(U)$ because composition of differentiable maps is differentiable. However, I have no clue in the other direction.
It's much easier now that you added finite dimensionality as a hypothesis.
Let $\pi_i\colon B\to\Bbb R$ (or $\Bbb C$) be projection onto the $i$th factor. Then you know that $f^*\pi_i = \pi_i\circ f$ is smooth for every $i$. It's immediate that that implies $f$ is smooth.