Does there exist a nonseparable Banach space $X$, a mapping $F: X\to X$, and an open nonempty subset $D\subset X$ such that $$ \forall\,E>0 \quad \exists\,\delta>0: \quad \forall\,x,y\in D \quad (0<\|x-y\|<\delta \Rightarrow \|F(x)-F(y)\|>E) \, ? $$
Of course, it is impossible if $X$ is separable.
It seems the following.
The answer is negative even when $D$ contains a complete metric space $C$ without isolated points, $X$ is a metric space and $F:D\to X$ is a map. Indeed, fix a point $y\in C$ and for each natural number $n$ put $C_n=\{x\in C: d(F(x), F(y))\le n\}$. Baire theorem implies that there exists a non-empty open subset $U$ of the space $C$ such that $U\subset \overline{C_n}$. Let $\delta>0$ be an arbitrary number. Since the set $C_n$ is dense in the open non-empty set $U$, there exist different points $x,x’\in U\cap C_n$ such then $d(x,x’)<\delta$. But $$|F(x)-F(x’)|\le |F(x)-F(y)|+|F(x’)-F(y)|\le 2n.$$