TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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The uniform limit of continuous real-valued functions on a metric space is continuous

Statement

Let $(X, d)$ be a metric space and let $f_{k} : X \to R$ be continuous for every $k \in N$ , where $R$ has its usual metric $d_{R} (s, t) = ∣ s - t ∣$ . If $f_{k} \to f$ uniformly on $X$ , then $f : X \to R$ is continuous.

Facts & Assumptions

Given: A metric space $(X, d)$ , continuous functions $f_{k} : X \to R$ , and uniform convergence $f_{k} \to f$ .

[A1]

Uniform convergence gives, for every real $η > 0$ , one index $N$ such that $∣ f_{k} (x) - f (x) ∣ < η$ for every $k \geq N$ and every $x \in X$ (Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions).

[A2]

Continuity of $f_{N}$ at $a$ means that for every real $η > 0$ there is $δ > 0$ such that $d (x, a) < δ$ implies $∣ f_{N} (x) - f_{N} (a) ∣ < η$ (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form, The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L1]

For reals $u, v$ , $∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣$ (The triangle inequality).

Proof

technique · direct

1.1A1choose

Fix $a \in X$ and a real $ε > 0$ . By uniform convergence choose $N$ such that $∣ f_{N} (x) - f (x) ∣ < ε / 3$ for every $x \in X$ .

1.2A2choose

By continuity of $f_{N}$ at $a$ , choose $δ > 0$ such that $d (x, a) < δ$ implies $∣ f_{N} (x) - f_{N} (a) ∣ < ε / 3$ .

2.1step 1.1step 1.2L1algebra

If $d (x, a) < δ$ , then $∣ f (x) - f (a) ∣ \leq ∣ f (x) - f_{N} (x) ∣ + ∣ f_{N} (x) - f_{N} (a) ∣ + ∣ f_{N} (a) - f (a) ∣ < ε$ .

3.1step 2.1A2∎

Thus $f$ is continuous at the arbitrary point $a$ , and hence continuous on $X$ .

The uniform limit of continuous real-valued functions on a metric space is continuous

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources