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

Throughout, $\mathbb{R}$ is the complete ordered field with its order and absolute value (Complete ordered field (least-upper-bound property), Basic properties of the absolute value), and neighbourhoods are those of The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$.

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

$(\mathrm{\forall }\epsilon >0)(\mathrm{\exists }\delta >0)(\mathrm{\forall }x\in A)[\mathrm{\mid }x-c\mathrm{\mid }<\delta ⟹\mathrm{\mid }f(x)-f(c)\mathrm{\mid }<\epsilon ],$

with $\epsilon$ and $\delta$ ranging over the positive reals. In the language of neighbourhoods: for every real $\epsilon >0$ there is a real $\delta >0$ with

$f(A\cap N_{\delta }(c))\subseteq N_{\epsilon }(f(c))\mathrm{.}$

$f$ is continuous on $A$ when it is continuous at every point of $A$.

The point $c$ is required to lie in $A$, and the condition is unpunctured. Both differ from The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$, and deliberately. There the quantifier runs over $0<\mathrm{\mid }x-c\mathrm{\mid }<\delta$, which removes $x=c$; here $x=c$ is allowed, and at $x=c$ the implication reads $\mathrm{\mid }f(c)-f(c)\mathrm{\mid }=0<\epsilon$, which is automatic. So allowing $x=c$ costs nothing, and it is what lets the definition be stated at every point of $A$, including the points where no limit exists.

Three clauses, and all three are part of the definition.

1. At a limit point. Suppose $c\in A$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$). Then $f$ is continuous at $c$ if and only if the limit of $f$ at $c$ exists and ${\lim }_{x\to c}f(x)=f(c)$ (The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$). Indeed, for a given $\epsilon >0$ a $\delta$ witnessing continuity witnesses the limit condition, because the limit condition quantifies over a subset of the points continuity quantifies over; and conversely a $\delta$ witnessing ${\lim }_{x\to c}f(x)=f(c)$ witnesses continuity, because the one point it omits, $x=c$, satisfies $\mathrm{\mid }f(c)-f(c)\mathrm{\mid }<\epsilon$ anyway.

2. At an isolated point. Suppose $c\in A$ is an isolated point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), so that $N_{\eta }(c)\cap A=\{c\}$ for some real $\eta >0$. Then every $f:A\to \mathbb{R}$ is continuous at $c$: take $\delta :=\eta$, so that the only $x\in A$ with $\mathrm{\mid }x-c\mathrm{\mid }<\delta$ is $c$ itself, and $\mathrm{\mid }f(c)-f(c)\mathrm{\mid }=0<\epsilon$.

3. On a set. Continuity on $A$ is continuity at each point of $A$, and nothing more. It is not a condition relating $f$ to points outside $A$.

Every point of $A$ is either a limit point of $A$ or an isolated point of $A$, and never both (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), so clauses 1 and 2 between them describe continuity at every point of $A$.

This is not the raw $\epsilon$-$\delta$ formula of FALSE: a function has at most one limit at every point of its domain, isolated points included. That item records what goes wrong when the punctured formula of The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$ is written down at an arbitrary point of the domain: at an isolated point it is satisfied vacuously by every real $L$ at once, so it defines nothing, and this library therefore leaves ${\lim }_{x\to c}f(x)$ undefined at an isolated point. Continuity at an isolated point is a different matter: the formula above is not vacuous — it is a genuine condition on $f(c)$, satisfied because $f(c)$ is the only value being compared with itself — and it names a single, well-defined property. The limit is undefined there; the continuity is defined, and is automatic. Clause 1 is the only place where the two notions meet, and it is stated only where the limit exists as a notion.

Where the distinction disappears. If $A$ is an open subset of $\mathbb{R}$ (Open subset of $\mathbb{R}$ (every point has a neighbourhood inside it), closed subset (complement open), and clopen), then every $c\in A$ has some $N_{\eta }(c)\subseteq A$, and a punctured neighbourhood is never empty (The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$), so every point of $A$ is a limit point of $A$ and clause 1 covers the whole of $A$. The same holds when $A$ is a nondegenerate interval (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length). Isolated points are what force clause 2 to exist at all, and they occur as soon as $A$ is allowed to be an arbitrary subset of $\mathbb{R}$, as in $A=\{0\}\cup [1,2]$.

• Continuity is local. If $\eta >0$ and $g:A\to \mathbb{R}$ agrees with $f$ on $A\cap N_{\eta }(c)$, then $f$ is continuous at $c$ if and only if $g$ is: any $\delta$ may be replaced by $\min \{\delta ,\eta \}$, after which the condition only ever evaluates the two functions where they agree. So continuity at $c$ sees only an arbitrarily small neighbourhood of $c$, exactly as the limit does (The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$, and passes to any subset of the domain having $c$ as a limit point).

• Continuity passes to subsets of the domain. If $B\subseteq A$ and $c\in B$, then continuity of $f$ at $c$ gives continuity of the restriction $f{\mathrm{\mid }}_B$ at $c$, with the same $\delta$: the condition on $f{\mathrm{\mid }}_B$ quantifies over fewer points. The converse fails, and the standard witness is the indicator of $\mathbb{Q}$ restricted to $\mathbb{Q}$, which is constant and hence continuous, while the indicator itself is continuous nowhere (The indicator of $\mathbb{Q}$ is continuous at no point of $\mathbb{R}$ ↗).

• The radius is a real number. As in The $\epsilon$-neighbourhood and the punctured $\epsilon$-neighbourhood of a point of $\mathbb{R}$, $\epsilon$ and $\delta$ range over the positive reals here. Restricting either quantifier to the positive rationals defines the same relation, by the passage recorded in The $\epsilon$-$\delta$ limit ${\lim }_{x\to c}f(x)=L$ of $f:A\to \mathbb{R}$ at a limit point $c$ of $A$: below every positive real lies a positive rational (The rationals embed densely in the reals), and a real $\delta$ may be shrunk to a rational one below it.

• The word continuous is used for two things in this library, and they agree. Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form defines continuity of a map between metric spaces, and $A\subseteq \mathbb{R}$ carries the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$. The two notions coincide, and that is proved, not assumed: Dictionary: for $A\subseteq \mathbb{R}$ with the metric $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$, continuity and uniform continuity of $f:A\to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace is the dictionary, and it is stated on this page precisely so that no later item has to guess.