On the existence of a non-negative function on a Banach space whose limit at every point is infinity.

Question

Does there exist a Banach space $ X $ (possibly non-separable) and a mapping $ F: X \to X $ such that $$ \forall a \in X: \quad \lim_{\substack{x \in X \setminus \{ a \} \\ x \to a}} \| F(x) \|_{X} = \infty? $$ Note: If the Banach space $ X $ is trivial, then the answer is a vacuous affirmative as the zero element, $ 0_{X} $, is the sole element of $ X $ and hence not a limit point. We may thus restrict our attention to only non-trivial Banach spaces.

Answer 1

No. Since the constant sequence $x_i = a$ converges to $a$ this would mean $\|F(a)\| = +\infty$.

However, as Berci suggested, you can ask for $f : X \to \mathbb R$, such that $$\limsup_{x \to a} f(x) = +\infty.$$ This is easy to accomplish, even with $X = \mathbb R$. Enumerate the rationals $\mathbb Q = \{q_i\}$ and define $$f(x) = \begin{cases} i & \text{if }x = q_i \text{ for some } i,\\0 & \text{if } x \not\in \mathbb Q.\end{cases}$$ This should do the job.