Weierstrass Function is Continuous

Theorem
Let $a \in \openint 0 1$.

Let $b$ be a strictly positive odd integer such that:


 * $\ds a b > 1 + \frac 3 2 \pi$

Let $f: \R \to \R$ be a real function defined by:


 * $\ds \map f x = \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$

for each $x \in \R$.

Then $f$ is well-defined and continuous.

Proof
Note that:


 * $\ds \sup_{x \mathop \in \R} \size {a^n \map \cos {b^n \pi x} } = a^n$

Since $a \in \openint 0 1$:


 * $\ds \sum_{n \mathop = 0}^\infty a^n$ converges.

So, by the Weierstrass M-Test:


 * $\ds \sum_{n \mathop = 0}^\infty a^n \map \cos {b^n \pi x}$ converges uniformly on $\R$.

That is, $f$ is well-defined.

Further, from Uniformly Convergent Series of Continuous Functions Converges to Continuous Function: Corollary:


 * $f$ is continuous.