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I´m basically trying to prove that if $S$=$\{(x,y) \in\mathbb R^2\mid y<x^2\}$. Then, for any $(x,y) \in S$ there exists a radius $\delta$ such that $B_\delta(x,y) \subseteq S$.

What values of $\delta$ would you recommend?

Weierstraß Ramirez
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For $\delta$ you could use something that is smaller than minimal distance between given point $p=(x,y)$ and the set boundary $\mathcal{B}$ (in your case it is $\{(x,y) | y = x^2$}). To find this distance you should find the line $l$ such that $l$ orthogonal to tangent line to $\mathcal{B}$ and $p \in l$. After that you get $p$ and $p' = l \cap \mathcal{B}$. Then you get distance between these points and use as delta something smaller. Distance can be simple Euclidean norm because all norms are equivalent in $\mathbb{R}^n$.

Artem
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Any disk will do provided it stays inside $S$. It is obviously possible to find such a disk, because the point must be a non-zero distance from the boundary.

almagest
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    I'd like to see this answer get marked in an exam. – Git Gud Sep 08 '14 at 21:54
  • That depends on the examiner. Have you read Littlewood's splendid debunking of the Maths Tripos' exam at Cambridge in his "Miscellany". He strongly attacks fake rigour. Anyone competent can translate the above into a totally rigorous argument. But the above is much clearer than that argument would be, because it is obvious at a glance. – almagest Sep 08 '14 at 22:04