$X \subseteq \Bbb R^m$ and $Y \subseteq \Bbb R^n$
Definition The map $f:X \to Y$ is $C^∞$ if for every $p∈X$ ,$\exists$ a nhbd $U_p$ of $p \in \Bbb R_n$ and a $C^∞$ map $g_p:U_p→R^m$ such that $g_p=f$ on $U_p ∩ X$. Let $U$ and $V$ are open sets in $\Bbb R_n$.
Definition Diffeomorphism: We say a function $h : U → V$ is diffeomorphism if it one-one, onto, and both $h, h^{−1} : V → U$ are $C^∞$ map.
Find a $C^∞$ map which is a one-one onto, inverse is continous but not a diffeomorphism.
Take $X = Y = \Bbb R$, and let $f:X\to Y$ be given by $f(x) = x^3$. Then $f$ is a $C^\infty$ bijection with continuous inverse $f^{-1}(y) = \sqrt[3]y$. However, the derivative of $f^{-1}$ fails to exist at $0$, so $f^{-1}$ is not $C^\infty$.