Define $B$ to be the set of functions $f:[0,1]\rightarrow \mathbb{R}$ for which there exists a dense set $C\subset [0,1]$ of computables numbers and an algorithm $F$ such that for any $x_0\in C,$ in the input $(f,x_0),$ $F$ return $(a_0,a_1,a_2,\ldots),$ where $a_n$ is the $n$th Taylor coefficient of $f$ around $x_0$ and for any $n,$ $a_n$ is a computable number.

Is it true that for any metric $d$ (non trivial) on the set of real analytic functions on $[0,1]$, any $f$ in this set and any $\epsilon>0$ there exists a $g\in B$ with $d(f,g)<\epsilon?$