$f:A\to \mathbb{R}$ is continuous at $c\in A$ if and only if ${\omega }_f(c)=0$

Let $A\subseteq \mathbb{R}$, let $f:A\to \mathbb{R}$ and let $c\in A$. Then

$f\text{ is continuous at }c⟺{\omega }_f(c)=0$

(Continuity of $f:A\to \mathbb{R}$ at a point of $A$ and on $A$: the $\epsilon$-$\delta$ condition, its agreement with ${\lim }_{x\to c}f(x)=f(c)$ at a limit point, and continuity at an isolated point, The oscillation ${\omega }_f(S)=\sup \{\mathrm{\mid }f(x)-f(y)\mathrm{\mid }:x,y\in S\}$ 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).

Since ${\omega }_f(c)\ge 0$ always (The oscillation ${\omega }_f(S)=\sup \{\mathrm{\mid }f(x)-f(y)\mathrm{\mid }:x,y\in S\}$ 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), the equivalent form of the right-hand side is: for every real $\epsilon >0$ there is a real $\delta >0$ with ${\omega }_f(A\cap N_{\delta }(c))<\epsilon$.

This is the tool that converts a pointwise condition into a set condition. Continuity at $c$ is a statement about $f$ near $c$ with a quantifier over $\epsilon$; ${\omega }_f(c)=0$ is the vanishing of a single extended real attached to the point. The change of form is what makes the discontinuity set accessible: the sets $\{x:{\omega }_f(x)\ge \epsilon \}$ are closed (For every real $\epsilon >0$ the set $\{x\in A:{\omega }_f(x)\ge \epsilon \}$ is the intersection with $A$ of a closed subset of $\mathbb{R}$; in particular it is closed in $\mathbb{R}$ when $A=\mathbb{R}$) and their union over $\epsilon =1,1\mathrm{/}2,1\mathrm{/}3,\dots$ is the discontinuity set (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).

• Where the extended reals are used, and where they are not. The definition of ${\omega }_f$ needs them, because $f$ may be unbounded near $c$ and the supremum may then be $+\mathrm{\infty }$; the proof needs only the order relations, and the two directions never compare an infinite value with a real except through the inequality $0\le {\omega }_f(c)\le \epsilon$, which already forces ${\omega }_f(c)$ to be real.

• The oscillation measures how badly continuity fails, not merely whether it does. The theorem uses only whether ${\omega }_f(c)$ vanishes, but the number itself carries more: the oscillation of Thomae's function at a point $c$ is proved below to be exactly $t(c)$ (The Dirichlet function is continuous at no point of $\mathbb{R}$, and Thomae's function is continuous at every irrational and at no rational, so its set of continuity points is exactly the set of irrationals and its oscillation at $c$ equals $t(c)$), so it is $1\mathrm{/}q$ at a rational with least denominator $q$ and $0$ at every irrational.