Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a convex function.
Then, is $f$ Borel-measurable?
From basic analysis, I know that $f$ is measurable (moreover, continuous) if $n=1$.
However, I am curious if this is still measurable if $n>1$.
2026-04-25 10:31:15.1777113075
Are convex functions measurable?
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Yes; convex functions on finite-dimensional spaces are continuous on the interior of their domain, hence they are Borel-measurable when defined over the entire space.