Non-density of restricted rationals (Q2276706)

The author introduced earlier restricted rational functions of the form p/q, where \(0<\mu (x)\leq q(x)\leq v(x)\) and \(\mu\) and \(\nu\) are continuous and fixed in C[\(\alpha\),\(\beta\) ]. If a degree of p is free, then p/q can approximate any \(f>0\) arbitrarily well (this property is called ``density of rationals in [\(\alpha\),\(\beta\) ]''), but it is not true if deg \(p\leq n\) (restricted rationals are non-dense). An open question arises: which functions can be as closely approximated by restricted rationals as by non-restricted rationals?