# Show that non-singular is necessary for the Inverse Function Theorem

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.