For $f : A \to \mathbb{R}$ the set of points of $A$ at which $f$ is discontinuous is the intersection with $A$ of an $F_\sigma$ subset of $\mathbb{R}$, and the set of points at which $f$ is continuous is the intersection with $A$ of a $G_\delta$ subset; for $A = \mathbb{R}$ the two sets are $F_\sigma$ and $G_\delta$ outright

Let $A \subseteq \mathbb{R}$ and let $f : A \to \mathbb{R}$. Write

$D \;:=\; \{\, x \in A : f \text{ is discontinuous at } x \,\}, \qquad C \;:=\; A \setminus D$

(Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point, Discontinuity of $f$ at a point of its domain, and its classification: removable discontinuity, jump discontinuity and essential discontinuity, equivalently Rudin's discontinuities of the first and of the second kind). Then:

1. Pointwise exhaustion. $D = \{\, x \in A : \omega_f(x) > 0 \,\}$ (The oscillation $\omega_f(S) = \sup\{\,|f(x) - f(y)| : x, y \in S\,\}$ of $f$ on a set and the oscillation $\omega_f(c) = \inf_{\delta > 0} \omega_f(A \cap N_\delta(c))$ at a point, both taken in the extended reals), and $D$ is the union of the increasing sequence of superlevel sets $D \;=\; \bigcup_{n \in \mathbb{N}} \{\, x \in A : \omega_f(x) \ge 1/\iota(n+1) \,\}$ (The canonical natural $\iota(n) = n \cdot 1_F$ of a field), whose thresholds are $1, 1/2, 1/3, \dots$.
2. Descriptive form. There is an $F_\sigma$ set $F \subseteq \mathbb{R}$ and a $G_\delta$ set $V \subseteq \mathbb{R}$ ($F_\sigma$ and $G_\delta$ subsets of $\mathbb{R}$) with $D \;=\; A \cap F, \qquad C \;=\; A \cap V, \qquad V = \mathbb{R} \setminus F ,$ and $F$ may be taken to be $\bigcup_{n \in \mathbb{N}} G_n$ with each $G_n$ a closed subset of $\mathbb{R}$ cutting down on $A$ to the $n$-th set of claim 1 (For every real $\varepsilon > 0$ the set $\{\,x \in A : \omega_f(x) \ge \varepsilon\,\}$ is the intersection with $A$ of a closed subset of $\mathbb{R}$; in particular it is closed in $\mathbb{R}$ when $A = \mathbb{R}$).

In particular, when $A = \mathbb{R}$ the discontinuity set $D$ is an $F_\sigma$ subset of $\mathbb{R}$ and the continuity set $C$ is a $G_\delta$ subset, and claim 1 reads $D = \bigcup_{n} \{\, x \in \mathbb{R} : \omega_f(x) \ge 1/\iota(n+1) \,\}$.

Claim 1 is stated separately because it is what is cited downstream. The exhaustion of $D$ by the superlevel sets $\{\omega_f \ge 1/\iota(n+1)\}$ is used directly wherever a property has to be established one threshold at a time — Baire's theorem: a Baire class one function on a closed bounded interval $[a,b]$ is continuous at the points of a dense subset of $[a,b]$ that is the trace of a $G_\delta$ set, so its set of discontinuities is meager shows each superlevel set nowhere dense and concludes that $D$ is meager — and that use needs the identity itself, not only the descriptive conclusion of claim 2.

The statement is relative on purpose. For a general domain $A$ the sets $D$ and $C$ are subsets of $A$, and neither is $F_\sigma$ or $G_\delta$ in $\mathbb{R}$ in general; what the proof produces are two subsets of $\mathbb{R}$ that cut down to them. The absolute form is stated only for $A = \mathbb{R}$, which is the case Every $G_\delta$ subset of $\mathbb{R}$ is the set of continuity points of some $f : \mathbb{R} \to \mathbb{R}$, so the $G_\delta$ sets are exactly the continuity sets and No function $\mathbb{R} \to \mathbb{R}$ is continuous at every rational and discontinuous at every irrational, because $\mathbb{Q}$ is not $G_\delta$ use.

• The exhaustion is indexed from $0$ and the thresholds are $1/\iota(n+1)$. $\mathbb{N}$ contains $0$, so the sequence of thresholds is $1, 1/2, 1/3, \dots$ and never $1/\iota(0)$, which is not defined. Writing the union as $\bigcup_{n \ge 1} \{\omega_f \ge 1/\iota(n)\}$ names the same family; the form above is used because a sequence in this library is a function on $\mathbb{N}$.

• The converse holds and is proved separately. Every $G_\delta$ subset of $\mathbb{R}$ is the continuity set of some function $\mathbb{R} \to \mathbb{R}$ (Every $G_\delta$ subset of $\mathbb{R}$ is the set of continuity points of some $f : \mathbb{R} \to \mathbb{R}$, so the $G_\delta$ sets are exactly the continuity sets), so the two classes coincide exactly. What the present theorem contributes is the direction that constrains: no function can have a continuity set that fails to be $G_\delta$, and No function $\mathbb{R} \to \mathbb{R}$ is continuous at every rational and discontinuous at every irrational, because $\mathbb{Q}$ is not $G_\delta$ spends that on $\mathbb{Q}$.