A separate continuous function not feebly continuous

We construct a separate continuous function $f\colon\mathbb{Q}\times\mathbb{Q}\to[0,1]$ and a dense subset $D\subseteq \mathbb{Q}\times\mathbb{Q}$ such that $f[D]$ is not dense in $f[\mathbb{Q}\times\mathbb{Q}]$, in other words, $f$ is separate continuous and not feebly (somewhat) continuous.