Analysis Problem of the Day 22

Today’s problem is Problem 5 from Stanford’s Fall 2024 Analysis Qual:

Problem 5: Let $f:[0,1] \to [0,\infty]$ be such that $\liminf_{x \to y} f(x) > f(y)$ for all $y$ such that $f(y) < \infty.$ Show that $S=\{x \in [0,1]: f(x)<\infty\}$ is at most countable.

Solution: This problem is similar to another problem from Stanford’s qual asking to show that a certain set is at most countable, and ultimately, utilizes the same idea with isolated points. Note that if $f(y)<\infty,$ then $f(y)$ is an isolated point in $f(U)$ for a small enough neighborhood $U$ of $y.$ Taking $\mathcal{B}$ to be a topological basis for $[0,1],$ it follows that $S$ is contained in the union of the sets of isolated points of all $U \in \mathcal{B}.$ Since $[0,1]$ is second countable, $\mathcal{B}$ is countable, and the set of isolated points of $f(U)$ is at most countable for any $U,$ so $S$ is at most countable as a countable union of at most countable sets.

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