# Proof about Borel measurable function

Let assume that the $$X = [x_{1,1},x_{1,2}]\times[x_{2,1},x_{3,2}]\times[x_{3,1},x_{3,2}]\times[x_{4,1},x_{4,2}]$$, $$Y=[y_{1,1},y_{1,2}]$$ where $$x_{n,m}$$ and $$y_{n,m}$$ are real numbers for all $$n,m$$.

Show that $$f: X \to Y$$ is Borel measurable function if $$f$$ is continuous.

I think the domain of the function $$f$$ is a Borel set because the $$X$$ is a cartesian product of Borel sets, Cartesian Product of Borel Sets is Borel Again.

And I know that continuous functions are measurable w.r.t. Borel sigma-algebra. But I want to clear proof for the problem.