FALSE: a function has at most one limit at every point of its domain, isolated points included

False claim: for every $A \subseteq \mathbb{R}$, every $f : A \to \mathbb{R}$ and every $c \in A$, at most one real $L$ satisfies

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

Read the claim carefully: it is about the raw formula $(\ast)$, extended to an arbitrary point $c$ of the domain. It is not a claim about The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$. That definition imposes $(\ast)$ only when $c$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and there at most one $L$ does satisfy it — that is exactly At a limit point of the domain a function has at most one limit, which is true and proved. The false claim is what one gets by deleting the limit-point requirement.

At an isolated point of $A$ the symbol $\lim_{x \to c} f(x)$ is undefined in this library, and the refutation below is the reason. If $c \in A$ is not a limit point of $A$ then some punctured neighbourhood of $c$ misses $A$ entirely (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$); the implication inside $(\ast)$ then has no instances at all for that $\delta$, so it holds vacuously, and it holds for every real $L$ at once. A formula satisfied by every real determines nothing, so no notation is introduced for it.

• This is the precise reason The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ carries the limit-point hypothesis. The hypothesis is not a convenience: it is what makes the quantified set $A \cap N^{*}_{\delta}(c)$ nonempty for every $\delta$, and hence what makes $(\ast)$ capable of pinning $L$ down. With it, At a limit point of the domain a function has at most one limit proves uniqueness; without it, uniqueness is simply false, as above.

• The true statement in the neighbourhood of the false one. For $c$ a limit point of $A$: at most one $L$ satisfies $(\ast)$ — At a limit point of the domain a function has at most one limit. For $c$ an isolated point of $A$: every $L$ satisfies $(\ast)$, by the argument of step 2.1, which uses nothing about $f$. So the dichotomy is total, and there is no intermediate case, because for $c \in A$ being isolated and being a limit point are exact opposites (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$).

• Some texts do define the limit at an isolated point, declaring it to be $f(c)$ by fiat, so that "limit" and "continuity" coincide on such points. That is a convention, not a theorem, and this library declines it: a convention that assigns a value to an expression which the definition leaves underdetermined would have to be carried, and checked, in every later statement about limits. The companion page's counterexample exhibits the underdetermination concretely.