1
$\begingroup$

I want to show that the following function is Borel measurable. Consider: $$f: \mathbb{R}^2 \rightarrow \mathbb{R}:\left\{ \begin{array}{ll} \sin\left(\frac{1}{x-y}\right) & x> y \\ x^2 + y^2 & x\leq y. \\ \end{array} \right. $$

So I think one can prove it by looking if the function is continuous (so measurable) or proving it by the defintition of a Borel measurable function. By proving it from the defenition, one can use that $\{x:f(x)>a\}$ is a Borel set. But I don't know how to proceed with this defenition.

$\endgroup$

closed as off-topic by Nosrati, Paul Frost, Cesareo, José Carlos Santos, Namaste Jan 6 at 20:54

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Nosrati, Paul Frost, Cesareo, José Carlos Santos, Namaste
If this question can be reworded to fit the rules in the help center, please edit the question.