Example of a map which is not a diffeomorphism

Question

Can anyone think of a bijective smooth map from a compact space to a huasdorff space which is not a diffeomorphism?

thanks

Answer 1

Sure, the standard example. $f\colon [-1,1]\to [-1,1]$, $f(x)=x^3$.

Answer 2

Take the identity map on $S^1$, pick a point, and add a "kink" so that the identity looks locally at that point like $x\mapsto x^3.$