the question can be found here, but is also repeated below.
Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is a function.
(a) For $k\in\mathbb{Z}^{+}$, let
$$G_k = \{a\in\mathbb{R}: \exists\delta>0 \text{ s.t. } \lvert f(b) – f(c)\rvert < \frac{1}{k}, \forall b,c\in(a-\delta, a+\delta\}.$$
Prove that $G_k$ is an open subset of $\mathbb{R}$ for each $k\in\mathbb{Z}^{+}$.
(b) Prove the set of points where $f$ is continuous equals $\cap_{k=1}^{\infty}G_{k}$.
(c) Conclude that the set of points at which $f$ is continuous is a Borel set.
So, I'm really just looking for a help at where to start. I think that I have an idea of where to set for (b) since the definition of $G_k$ is essentially just the definition of continuity using $\varepsilon_k = \frac{1}{k}$, but I'm fairly lost on what to do for the other sections.
Thank you!
Best Answer
Welcome to MSE!
Hint:
For (a), what happens if we wiggle $a$ a little bit? That is, we know that $a \in G_k$ satisfies some property. Can you show that actually some open set around $a$ is also in $G_k$? Do you see why that would show that $G_k$ is open?
As for showing it, we know that if we stay within $\delta$ of $a$, something good happens. Can you show that for every point within $\delta/2$ of $a$ (say), there's a smaller radius around those points for which good things keep happening?
For (b), you're completely right!
For (c), you've shown the points of continuity is a (countable!) intersection of open sets $G_k$. Why must this be borel?
I hope this helps ^_^