The $\varepsilon$-$\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$

Throughout, $\mathbb{R}$ is the complete ordered field (Complete ordered field (least-upper-bound property)) with its order and absolute value (Order on the reals).

Let $A \subseteq \mathbb{R}$, let $f : A \to \mathbb{R}$, let $c \in \mathbb{R}$ be a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and let $L \in \mathbb{R}$. We say that $f(x)$ tends to $L$ as $x$ tends to $c$, and write

$\lim_{x \to c} f(x) = L ,$

when

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

where $\varepsilon$ and $\delta$ range over the positive reals.

In the language of neighbourhoods (The $\varepsilon$-neighbourhood and the punctured $\varepsilon$-neighbourhood of a point of $\mathbb{R}$) the condition reads: for every real $\varepsilon > 0$ there is a real $\delta > 0$ with

$f\bigl(A \cap N^{*}_{\delta}(c)\bigr) \;\subseteq\; N_{\varepsilon}(L),$

$N^{*}_{\delta}(c) = \{\, y : 0 < |y - c| < \delta \,\}$ being the punctured $\delta$-neighbourhood of $c$ and $N_{\varepsilon}(L) = (L - \varepsilon,\ L + \varepsilon)$ the open interval of Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length. The two forms agree because $|f(x) - L| < \varepsilon$ says exactly $f(x) \in N_\varepsilon(L)$, and $0 < |x - c| < \delta$ says exactly $x \in N^{*}_\delta(c)$.

Three features of this definition are load bearing, not decoration.

1. $c$ is required to be a limit point of $A$. By Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ that says every punctured neighbourhood of $c$ meets $A$, so for every $\delta > 0$ the set $A \cap N^{*}_\delta(c)$ over which the implication quantifies is nonempty. Drop the requirement and the implication can be satisfied vacuously by every real $L$ at once, which is exactly what FALSE: a function has at most one limit at every point of its domain, isolated points included records. At a point of $A$ that is not a limit point of $A$ — an isolated point — the symbol $\lim_{x \to c} f(x)$ is therefore not defined in this library.

2. $c \in A$ is not required. A limit point of $A$ need not belong to $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$), and the definition never evaluates $f$ at $c$. This is what allows a limit to be taken at a point where the function is not defined at all, as at $0$ for $x \mapsto x\,\psi(1/x)$.

3. The value $f(c)$, when it exists, is irrelevant. The hypothesis $0 < |x - c|$ excludes $x = c$ from the quantifier, so changing $f$ at the single point $c$ changes nothing. Equality of the limit with the value is an extra condition, not a consequence: FALSE: $\lim_{x \to c} f(x) = f(c)$ whenever both sides exist.

The notation presumes uniqueness. Writing $\lim_{x \to c} f(x) = L$ treats the left-hand side as a name for a single real number, which is legitimate only because at a limit point at most one $L$ can satisfy the displayed condition. That obligation is discharged by At a limit point of the domain a function has at most one limit ↗, recorded in this item's justified_by. As with $\sup S$ (Conventions: $\sup \emptyset$, unbounded sets, and the extended reals) and $\lim_k x_k$ (A sequence has at most one limit), the symbol is written only for a function already known to have a limit at $c$.

Real and rational $\varepsilon$ define the same relation. Above, $\varepsilon$ and $\delta$ range over the positive reals. Restricting either quantifier to the positive rationals gives the same relation: every positive rational is a positive real, and below every positive real lies a positive rational (The rationals embed densely in the reals), so an $\varepsilon$-condition verified for all positive rationals is verified for an arbitrary positive real $\eta$ by running it at a rational $\varepsilon$ with $0 < \varepsilon < \eta$, and a $\delta$ produced as a real may be shrunk to a rational one below it. This is the passage sanctioned in the remarks of Sequences of reals: bounded, eventually, frequently, tails, subsequences, and it is what lets this definition be compared with Limits and Cauchy sequences of reals, whose $\varepsilon$ is rational, in Heine criterion: $\lim_{x \to c} f(x) = L$ iff $f(x_k) \to L$ for every sequence in $A \setminus \{c\}$ converging to $c$.

• Terminology. Limit point here is a property of the set $A$ and the point $c$, in the sense of Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$; it has nothing to do with subsequential limits (Subsequential limit of a real sequence, and the subsequential limit set), and the distinction is the one that item records.

• Why the punctured condition, and not $|x - c| < \delta$. With the unpunctured condition the definition would force $f$ to be defined at $c$ and would force $|f(c) - L| < \varepsilon$ for every $\varepsilon$, that is, $L = f(c)$. The resulting notion is continuity at $c$, a strictly stronger condition, and conflating the two is the error catalogued in FALSE: $\lim_{x \to c} f(x) = f(c)$ whenever both sides exist.

• One-sided and infinite variants. Restricting the domain to one side of $c$ gives the one-sided limits of The left and right limits of $f$ at $c$, as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$; replacing the conditions on $x$ or on $f(x)$ by unboundedness conditions gives the limits at and to infinity of Limits at $+\infty$ and $-\infty$, and infinite limits at a point. Both are built on this definition rather than beside it.