Can you please give an example of a convex and closed function that is not continuous?
2026-04-03 14:12:52.1775225572
Continuity is not guaranteed even when the function is closed and convex. Why?
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Define $f \colon \mathbb R \to \mathbb R \cup \{+\infty\}$ via $$ f(x) = \begin{cases} 0 & \text{if } x = 0, \\ \infty & \text{else}.\end{cases}$$ Clearly, this function is closed and convex (its epigraph is closed and convex), but not continuous.