Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.

Let $f$ be a $m(X,X^*)$-continuous function on the subset $H$ of $X$ that goes to $\mathbb{R}$.

Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?

Let $X$ be a Banach space equipped with the Mackey topology $m(X,X^*)$. We suppose that $\big(X,m(X,X^*)\big)$ is separable space. Let $H$ be a countable, $m(X,X^*)$-dense subset with $(H=-H)$.

Let $f$ be a sublinear function $m(X,X^*)$-continuous on the subset $H$ of $X$ that goes to $\mathbb{R}$.

Can we say that $f$ admits a $m(X,X^*)$-continuous extension to $X$?