Let $f : R^n → R$ be a continuous function. Let $c ∈ Im f$. When is $f ^{−1}(c)$ a manifold?
I know that if $f$ is a smooth function then $f ^{−1}(c)$ is a manifold if $c$ is a regular value. But what can we say about continuous functions?
Also, if $g : R → R^n$ be a continuous function. When is $Im (g)$ a manifold?
You can't really say much - given an arbitrary closed set $A \subseteq \mathbb{R}^n$, the distance function $f(x) = d(A,x)$ is continuous and $A = f^{-1}(0)$ so any closed subset is the inverse image of a continuous function but most of them are very far from being manifolds.