Lebl-Basic-Analysis-I · Problem 4.4.5

The inverse function theorem guarantees $g$ exists and is differentiable, but you need to show smoothness propagates: if $f$ is $C^k$, so is $g$. The key insight is to use the implicit relationship $f(g(x)) = x$ and differentiate it repeatedly. Each differentiation produces a formula for the next derivative of $g$ in terms of derivatives of $f$ and lower derivatives of $g$—so by induction, if $f$ is $C^k$, then $g$'s derivatives up to order $k$ exist and are continuous.

The inverse function theorem guarantees $g$ exists and is differentiable, but you need to show smoothness propagates: if $f$ is $C^k$, so is $g$. The key insight is to use the implicit relationship $f(g(x)) = x$ and differentiate it repeatedly. Each differentiation produces a formula for the next derivative of $g$ in terms of derivatives of $f$ and lower derivatives of $g$—so by induction, if $f$ is $C^k$, then $g$'s derivatives up to order $k$ exist and are continuous.