Def 1. Let $f:I\to J$. The function $f(x)$ is $C^r$-diffeomorphism if $f(x)$ is a $C^r$-homeomorphism such that $f^{-1}(x)$ is also $C^r$.
Def 2. Let $f:I\to J$. The function $f(x)$ is homeomorphism if $f(x)$ is one-to-one, onto, and continuous, and $f^{-1}(x)$ is also continuous.
From these two definitions, can I say that $f(x)=\frac{1}{x}$ is a diffeomorphism (in its domain of definition)? I think yes, because: let $f:I\to J$, and $I=J=R-\{0\}$, then $f^{-1}(x)$ is continuous in $J=R-\{0\}$. What do you think?
Thanks