All Questions

Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in $...

Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$, such that for all $c\in \mathbb{R}$ both sets $$ f_{<c}~=~ f^{-1}\left( (-\...