# Showing that a multivariable function is Borel measurable. [closed]

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.

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

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