I try to understand the inverse function theorem by trying learn what will happen if one of conditions isn't satisfy. I try to find if it's possible to find an example such that
a function $f:[a,b] \to [a,b]$ is bijective,
$f$ is infinitely differentiable,
$f$ has some continuous inverse function $g$,
but $g$ is not differentiable at some point of $[a,b]$
I don't think there is an example satisfy all of above, but I'm not so sure about that.