# Find boundary of a set (interval), What to use?

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.

• Do you know what $f$ is? For example is it a continuous function at $0$? – AnyAD Apr 25 '17 at 10:23
• @Any I am able to put some conditions on $f$, so it is possible to suppose $f$ continuous. – Nizar Apr 25 '17 at 12:22
• Maybe you could use continuity of $f$ at zero, and find each of the two bounds by considering the given function as function of one variable (consider the bahaviour as one of the $x_i$s is fixed. – AnyAD Apr 25 '17 at 12:50
• @Any By fixing one variable, we will loose the possibility of having the same results when varying both variables. May be I misunderstand you, so can you elaborate more please? – Nizar Apr 25 '17 at 13:44