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.


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