$f$ admits a continuous extension to $X$?

Question

Let $(X,\|.\|)$ be a separable Banach space and $H$ be a countable, dense subset.

Let $f$ be a sublinear function continuous on the subset $H$ of $X$ that goes to $\mathbb{R}$, such that there exists a sublinear function $g:X\to \mathbb{R}$ continuous on $X$, such that: $$ |f(x)|\leq g(x)\qquad\forall x\in H ~~(*) $$ Can we say that $f$ admits a continuous extension to $X$?

An idea please

Answer 1

There is only way to define such extension, given such conditions.

Let $x\in X$ and consider $\{h_n\}\subset H$ with $h_n\to x$. Define $\tilde{f}(x)=\lim f(h_n)$.

Prove that $\tilde{f}\colon X\to\mathbb{R}$ is well defined (do not depend on the choice of the sequence $h_n$) and satisfies all your requirements.

Note that in order to $f$ be sublinear we must assume that $H$ is a subspace of $X$ (or at least assume that $f$ satisfies sublinearity conditions whenever they make sense).