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

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 $\varepsilon$-neighbourhood and the punctured $\varepsilon$-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

$(\forall \varepsilon > 0)\ (\exists \delta > 0)\ (\forall x \in A)\ \bigl[\ |x - c| < \delta \ \Longrightarrow\ |f(x) - f(c)| < \varepsilon\ \bigr],$

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

$f\bigl(A \cap N_{\delta}(c)\bigr) \;\subseteq\; N_{\varepsilon}\bigl(f(c)\bigr).$

$f$ 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 $\varepsilon$-$\delta$ 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 $0 < |x - c| < \delta$, which removes $x = c$; here $x = c$ is allowed, and at $x = c$ the implication reads $|f(c) - f(c)| = 0 < \varepsilon$, 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 $\varepsilon$-$\delta$ 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 $\varepsilon > 0$ 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 $|f(c) - f(c)| < \varepsilon$ 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 $|x - c| < \delta$ is $c$ itself, and $|f(c) - f(c)| = 0 < \varepsilon$.

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 $\varepsilon$-$\delta$ 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 $\varepsilon$-$\delta$ 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 $\varepsilon$-neighbourhood and the punctured $\varepsilon$-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 $f|_B$ at $c$, with the same $\delta$: the condition on $f|_B$ 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 $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$, $\varepsilon$ 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 $\varepsilon$-$\delta$ 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 $\varepsilon$-$\delta$ form defines continuity of a map between metric spaces, and $A \subseteq \mathbb{R}$ carries the metric $d(x,y) = |x-y|$. The two notions coincide, and that is proved, not assumed: Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, 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.