A counterexample for $H^{\infty}$ approximable functions

Let $\mathbb{D}$ be the unit disk. We show that for some relatively closed set $F\subset \mathbb{D}$ there is a function $f$ that can be uniformly approximated on $F$ by functions of $H^{\infty}$, but such that $f$ cannot be written as $f= h+g$, with $h\in H^{\infty}$ and $g$ uniformly continuous on $F$. This answers a question of Stray.