Let $f$ and $h$ be two lower semi-continuous functions from a metric space $X$ into a metric space $Y$. If $f$ and $h$ coincide in a dense subset of $X$, do $f$ and $h$ coincide on the whole space $X$?.
2026-04-09 17:06:54.1775754414
Lower semi-continuous functions coinciding on a dense subset
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No. For example, take a continuous function, and lessen the value of the function at a single point (well, a single non-isolated point!).