TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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The classical Weierstrass series converges uniformly to a continuous function

Statement

Let $0 < a < 1$ and let $b > 1$ be an odd integer, and let $W_{a, b}$ be the series of The classical Weierstrass function. The series defining $W_{a, b}$ converges absolutely at every real point and uniformly on $R$ .

Its sum $W_{a, b} : R \to R$ is continuous. If

$S_{N} (x) : = \sum_{n = 0}^{N} a^{n} \cos (b^{n} π x),$

then the partial sums converge uniformly to $W_{a, b}$ on $R$ .

Facts & Assumptions

Given: Parameters $0 < a < 1$ and an odd integer $b > 1$ , with summands $f_{n} (x) : = a^{n} \cos (b^{n} π x)$ and partial sums $S_{N}$ .

[L1]

For every real $x$ , $∣ \cos x ∣ \leq 1$ (Parity and the Pythagorean identity for sine and cosine).

[L2]

If $∣ r ∣ < 1$ , then the series $\sum_{k = 0}^{\infty} r^{k}$ converges (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L3]

If $∣ f_{k} (x) ∣ \leq M_{k}$ for all $k$ and $x$ , where the nonnegative scalar series $\sum M_{k}$ converges, then $\sum f_{k} (x)$ converges absolutely at every $x$ and the function series converges uniformly (The Weierstrass M-test gives absolute pointwise convergence and uniform convergence of a function series).

[L4]

The functions $\sin$ and $\cos$ are differentiable on $R$ (The derivatives of sine and cosine are cosine and minus sine).

[L5]

A differentiable real function is continuous at every point where it is differentiable (A function differentiable at $c$ is continuous at $c$ ).

[L6]

Sums and scalar multiples of continuous real functions are continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, claim 1).

[L7]

A composite of continuous maps is continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 1).

[L8]

A uniform limit of continuous real-valued functions on a metric space is continuous (The uniform limit of continuous real-valued functions on a metric space is continuous).

[L9]

Every polynomial function is continuous (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, claim 5).

Proof

technique · direct

1.1givenL1algebra

For every $n \in N$ and $x \in R$ , $∣ f_{n} (x) ∣ = a^{n} ∣ \cos (b^{n} π x) ∣ \leq a^{n}$ .

1.2givenL2

Since $0 < a < 1$ , the majorant series $\sum_{n = 0}^{\infty} a^{n}$ converges, including its first term $a^{0} = 1$ .

1.3L4L5L6L7L9

Cosine is continuous by [L4] and [L5]; the map $x \mapsto b^{n} π x$ is a polynomial and hence continuous by [L9], so each $f_{n}$ is continuous by [L7], and every finite partial sum $S_{N}$ is continuous by [L6].

2.1step 1.1step 1.2L3

Applying [L3] to steps 1.1 and 1.2 proves absolute convergence at every real point and uniform convergence of the partial sums to $W_{a, b}$ on $R$ .

3.1step 1.3step 2.1L8∎

The functions $S_{N}$ are continuous by step 1.3 and converge uniformly by step 2.1, so [L8] makes their sum $W_{a, b}$ continuous.

Depends on

Used by

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Sources

John K. Hunter, An Introduction to Real Analysis, Theorem 9.22 and Example 9.24 (standard reference, not scraped)
Jeff Calder, Weierstrass's Non-Differentiable Function, Theorem 1 (standard reference, not scraped)