A fairly general form of the Inverse Function Theorem is:
Suppose $X, Y$ are Banach spaces, $U \subset X$ is open and $f:U \to Y$ is continuously differentiable. If for some $x \in U$ the derivative $Df(x)$ is invertible, then there exists a neighborhood $V \subset f(U)$ such that $f(x) \in V$ and a continuously differentiable function $g: V \to U$ such that $f(g(x)) = x$ for all $x \in V$.
A question I have had is whether there are any sufficient conditions such that the converse holds, i.e., if $Df(x)$ is not invertible then $f$ is locally not invertible.?
Thanks in advance.