So directly I will State my problem:
Given $\epsilon >0$ and some function $f$, required to find $\epsilon_1, \epsilon_2 >0$ such that, for any $x_1, x_2 \in \mathbb{R}$ we have $$ |x_1|\leq \epsilon_1 \text{ and } |x_2|\leq \epsilon_2 \implies |f(x_1,x_2) |\le \epsilon $$
How I describe the problem: The idea is to find intervals ( or sets) in which the function $f$ on these two sets will not exceed a given threshold.
What I have tried: I tried to transform this into an optimization problem. What stopped me is that the boundary restrictions ( $\epsilon_1$ and $\epsilon_2$) are not constraints they are the variables that we are searching for.
My knowledge in optimisation is modest (I took 2 courses in optimisation three years ago) so I am asking for some help. May be there will be another way to do such problems, can some one help me please.