From the book Introduction to Differentiable Topology by TH Brocker and K Janich.
1.7 Definition: A diffeomorphism is an invertible differentiable map
Below:
A differentiable homeomorphism need not be a diffeomorphism, as is shown by the map $\mathbb{R} \rightarrow \mathbb{R}, x \rightarrow x^3$.
I don't understand why this map is not a diffeomorphism. This appears differentiable ($x \rightarrow 3x^2$) and invertible $x \rightarrow x^{1/3}$. So I don't understand this example. I'm also confused because "differentiable homeomorphism" seem to equate differentiable and invertible. I must be misunderstanding a concept.
Diffeomorphism requires both the function and its inverse are differentiable. But for $f(x)=x^3$, its inverse $f^{-1}(x)=\sqrt[3]{x}$ fails to be differentiable at $x=0$, so $f$ is not a diffeomorphism.