Relaxing continuous differentiability in the inverse function theorem

Question

Let $f$ be a continuous injective function on a ball centered at the origin of $\bf R^n$ into $\bf R^n$. Suppose that $f$ is differentiable at the origin with non-zero Jacobian determinant. Denote the inverse of $f$ by $g$. Must $g^\prime(0)$ exist and equal the inverse of the matrix $f^\prime(0)$? I know this is true if $n = 1$.

Answer 1

First note that by reducing the size of the ball and closing it, we obtain a compact set on which $f$ is continuous and injective. It follows that $f$ is a homeomorphism on this smaller closed ball. Thus, $f$ is a homeomorphism on the smaller open ball, so we may as well assume that $f$ is a homeomorphism on the original ball. WLOG we can also assume that $f(0) = 0$. Since $f^\prime(0)$ exists, we then have $f(x) = f^\prime(0) x + o(\|x\|)$. Using the homeomorphic properties of $f$ and the continuity of linear maps, we see that $f^\prime(0)^{-1} y = g(y) + o(\|y\|)$. It follows that $g^\prime(0) = f^\prime(0)^{-1}$.