Composition of smooth map

Question

In the context of differential manifold, can a smooth map composed with a non-smooth map be equal to a smooth map?

Thank you!

Answer 1

How about a constant function $g(x)=p$ composed with $f(x)=x^{\frac13}$ on $\Bbb R$? The composition $g\circ f$ would be constant, hence smooth.

Or let $g(x)=x^3$, for that matter.