Suppose $f$ is a polynomial of degree $d$ and $B\subset\mathbb{R}$ is an open interval. Then let $A=\{x\in B:|f(x)|<\epsilon\}$ then can we always choose $d+1$ points $x_i$ from the set $A$ i.e $|f(x_i)|<\epsilon$ such that $|x_i-x_j|\geq\frac{\sigma}{2d}$ for $1\leq i<j\leq d+1$( where $\sigma$ is the Lebesgue measure of the set $A$)
I was initially thinking to take the above points as roots of the polynomial but that doesn't necessarily belong to $B$ and if the roots are same then also it is difficult. Any idea regarding how to choose the points so that it would satisfy the conditions above? Any hints would be very much helpful.