In this question concerning the difference between homeomorphisms (continuous bijections with continuous inverses mapping between two spaces) and diffeomorphisms (bijections with smooth inverses mapping between two spaces), multiple answers give $f(x) = x^3$ as an example of a homeomorphism that is not a diffeomorphism.
I see that $f^{-1}(x) = x^{1/3}$ is not smooth at $x=0$. However, why can't we use $g(x) = x^2$ as an example of a homeomorphism that is not a diffeomorphism?
If the domain and codomain are (as in the linked examples) $\Bbb R$, then $f(x)=x^2$ is not a bijection.