A separate continuous function not feebly continuous

TL;DR

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Contribution

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Abstract

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.