ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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$x \cdot 1_{Q} (x)$ has a limit at $0$ and at no other point

Example

With $1_{Q}$ as in The indicator of $Q$ has a limit at no point of $R$ , let

$d : R \to R, d (x) : = x \cdot 1_{Q} (x),$

so $d (x) = x$ for rational $x$ and $d (x) = 0$ for irrational $x$ . Then the limit of $d$ at $0$ exists, with

$\lim_{x \to 0} d (x) = 0,$

and at every $c \neq 0$ the function $d$ has no limit.

The point of the example. The factor $1_{Q}$ has a limit nowhere; multiplying it by $x$ repairs exactly one point, and only that one. The repair at $0$ is the squeeze theorem (If $f \leq g \leq h$ near $c$ and $f$ and $h$ have the same limit at $c$ , then so does $g$ ) applied to $- ∣ x ∣ \leq d (x) \leq ∣ x ∣$ ; the failure elsewhere is the same two-sequence argument as in The indicator of $Q$ has a limit at no point of $R$ , now with image limits $c$ and $0$ , which are distinct precisely because $c \neq 0$ .

Facts & Assumptions

Given: The canonical copy $Q \subseteq R$ of the rationals, the irrationals $X = R ∖ Q$ , the function $d (x) = x \cdot 1_{Q} (x)$ , and a real $c \neq 0$ .

[L1]

The values of $d$ : $d (x) = x$ for $x \in Q$ and $d (x) = 0$ for $x \in X$ ; every real lies in exactly one of $Q$ and $X$ (The indicator of $Q$ has a limit at no point of $R$ ).

[L2]

The limit condition (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ): $\lim_{x \to c} f (x) = P$ means that for every real $ε > 0$ there is a real $δ > 0$ such that every $x$ in the domain with $0 < ∣ x - c ∣ < δ$ satisfies $∣ f (x) - P ∣ < ε$ .

[L3]

Squeeze theorem: if $f \leq g \leq k$ on $A \cap N_{η}^{*} (c)$ for some real $η > 0$ and the limits of $f$ and of $k$ at $c$ exist and are equal to $L$ , then the limit of $g$ at $c$ exists and equals $L$ (If $f \leq g \leq h$ near $c$ and $f$ and $h$ have the same limit at $c$ , then so does $g$ ).

[L4]

Nonexistence criterion (A function has no limit at $c$ as soon as two sequences in $A ∖ {c}$ tending to $c$ give different limits of the values).

[L5]

For every real $c$ there are a sequence $(q_{k})$ with all terms in $Q ∖ {c}$ and a sequence $(u_{k})$ with all terms in $X ∖ {c}$ , both converging to $c$ . This is exactly what is established in the course of The indicator of $Q$ has a limit at no point of $R$ , from density (Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable, The closure equals the set together with its limit points, equals the set of points every neighbourhood of which meets it, and is the smallest closed superset; a set is closed iff it contains its limit points, Interior, closure, boundary and exterior of a subset of $R$ , The rationals embed densely in the reals) and the sequential characterisation of the closure (A point lies in the closure of $A \subseteq R$ iff some sequence in $A$ converges to it, so a subset of $R$ is closed iff it is sequentially closed, Sequences of reals: bounded, eventually, frequently, tails, subsequences, Limits and Cauchy sequences of reals); the countable choice spent there is inherited here.

[L6]

Every real is a limit point of $R$ , punctured neighbourhoods being never empty (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ , The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ ).

[L7]

Absolute value: $∣ u ∣ \geq 0$ ; $- ∣ u ∣ \leq u \leq ∣ u ∣$ ; $∣ u ∣ = u$ for $u \geq 0$ ; $∣ 0 ∣ = 0$ (Basic properties of the absolute value). Order arithmetic: trichotomy and totality; $0 < 1$ , so $2 > 0$ and $ε / 2 < ε$ for $ε > 0$ (The multiplicative identity is positive, Order is preserved by adding a constant and by adding inequalities, Sign rules for products and monotonicity of multiplication, Ordered field).

[L8]

A constant sequence converges to its value (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

Verification

technique · direct

1.1

For every $x \in R$ , $- ∣ x ∣ \leq d (x) \leq ∣ x ∣$ : if $x \in Q$ then $d (x) = x$ and $- ∣ x ∣ \leq x \leq ∣ x ∣$ ; if $x \in X$ then $d (x) = 0$ and $- ∣ x ∣ \leq 0 \leq ∣ x ∣$ .

L1L7

1.2

Every real is a limit point of $R$ ; in particular $0$ and the given $c$ are.

1.3

The functions $x \mapsto - ∣ x ∣$ and $x \mapsto ∣ x ∣$ have limit $0$ at $0$ : given a real $ε > 0$ take $δ : = ε$ ; every $x$ with $0 < ∣ x - 0 ∣ < δ$ satisfies $∣ ∣ x ∣ - 0 ∣ = ∣ x ∣ < ε$ and $∣ - ∣ x ∣ - 0 ∣ = ∣ x ∣ < ε$ .

L2L7

2.1

The three functions satisfy $- ∣ x ∣ \leq d (x) \leq ∣ x ∣$ on all of $R$ , in particular on $R \cap N_{1}^{*} (0)$ , and the outer two have limit $0$ at $0$ ; since $0$ is a limit point of $R$ , the squeeze theorem [L3] gives that the limit of $d$ at $0$ exists and equals $0$ .

step 1.1step 1.2step 1.3L3

2.2

Fix the real $c \neq 0$ . By [L5] there are a sequence $(q_{k})$ with all terms in $Q ∖ {c}$ and a sequence $(u_{k})$ with all terms in $X ∖ {c}$ , both converging to $c$ .

step 1.2L5choose

3.1

By [L1], $d (q_{k}) = q_{k}$ for every $k$ , so the image sequence $(d (q_{k}))$ is $(q_{k})$ itself and converges to $c$ ; and $d (u_{k}) = 0$ for every $k$ , so that image sequence is constant and converges to $0$ . Since $c \neq 0$ , the two limits are distinct, and both sequences have all their terms in $R ∖ {c}$ and converge to $c$ ; by [L4] the function $d$ has no limit at $c$ .

step 2.2L1L4L7L8

4.1

So the limit of $d$ exists at $0$ , with value $0$ , and fails to exist at every other real: $d$ has a limit at exactly one point.

step 2.1step 3.1∎

Remarks

Why $0$ is the exceptional point. The squeeze bound $∣ d (x) ∣ \leq ∣ x ∣$ is useful only where $∣ x ∣$ is small, that is near $0$ ; at any other $c$ the two bounding functions have limit $∣ c ∣ \neq 0$ and $- ∣ c ∣$ , which are different, so the squeeze theorem says nothing there. That is not an accident of the proof: the two-sequence argument shows the limit genuinely fails at every such $c$ .
The value $d (0) = 0$ happens to equal the limit, since $0$ is rational, so $d$ satisfies at $0$ the equality that FALSE: $\lim_{x \to c} f (x) = f (c)$ whenever both sides exist shows is not automatic. It is the only point of $R$ at which $d$ does so.
Contrast with $ψ (1 / x)$ . There the oscillation is bounded and the failure is confined to a single point, $0$ , with the multiplication by $x$ repairing precisely that point ( $x ψ (1 / x) \to 0$ as $x \to 0$ , by the squeeze theorem). Here the failure is everywhere and the multiplication repairs precisely one point. The two examples are the same mechanism — a bounded factor damped by a vanishing one — applied to opposite kinds of irregularity.

Depends on

Used by

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Sources

Dirichlet function (Wikipedia) (standard reference, not scraped)
Squeeze theorem (Wikipedia) (standard reference, not scraped)
J. Lebl, Basic Analysis I, §3.1 (standard reference, not scraped)

x⋅1Q(x)x⋅1Q​(x) has a limit at 00 and at no other point