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Suppose $f: \mathbb{R} \to \mathbb{R}$ is a function. For $k \in \mathbb{Z}^+$, let $$G_k = \{a\in\mathbb{R}: \textrm{there exists $\delta > 0$ such that $|f(b) - f(c)| < \frac{1}{k}$ for all $b,c, \in (a-\delta, a+\delta)$}\}.$$ Prove that $G_k$ is an open subset of $\mathbb{R}$ for each $k \in \mathbb{Z}^+$.

I've observed that each $a \in G_k$ is contained in an open interval $(a-\delta, a+\delta)$ for some $\delta$, but I am not sure how to show that $G_k$ is an open subset for EACH $k \in \mathbb{Z}^+$. I am trying to show that each $a$ is contained in an open cube contained in $G_k$ which is sufficient enough to conclude that $G_k$ is open, but how can I connect all of this together? Am I on the right track or am I overlooking something?

Evan Kim
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  • What is your $f$ continuous or not? – Sumanta Sep 10 '20 at 20:05
  • THe goal of the exercise is to show that if $f: \mathbb{R} \to \mathbb{R}$, then the set of points at which $f$ is continuous is the intersection of a sequence of open sets and thus is a Borel set. – Evan Kim Sep 10 '20 at 20:15
  • See also https://math.stackexchange.com/questions/67620/set-of-continuity-points-of-a-real-function and https://math.stackexchange.com/questions/513339/the-set-of-points-at-which-a-real-function-is-continuous-is-borel – Sumanta Sep 10 '20 at 20:21
  • yes thank you! That is helpful – Evan Kim Sep 10 '20 at 20:30

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