I am attempting the following problem:
Show that the condition that $dF(a)$ be non-singular is necessary in the inverse function theorem by showing that if a function $F$ from a neighborhood of $a$ in $\mathbb{R}^p$ to $\mathbb{R}^p$ is differentiable at $a$ and has an inverse function at $a$ which is differentiable at $F(a)$, then $dF(a)$ is non-singular.
I know that for some neighborhood $V$ of $a$, $F^{-1}$ is smooth on $W=F(V)$ and $F$ is one-to-one from $V$ onto $W$ and $F^{-1}(u) = x$ if $F(x) = u$.
I am not sure how to approach this problem. I don't feel like I have an intuitive grasp of what I am trying to show, and reading through theorems hasn't given me any insight into what approach might be successful.
HINT: Write down what your hypotheses are and use the chain rule.