# How to prove using elementary methods that this function is everywhere continuous but nowhere differentiable?

Let $$f$$ be the function defined on all of $$\mathbb{R}$$ by the formula $$f(x) \colon= \sum_{n=0}^\infty \frac{1}{2^n} \cos \left( 3^n x \right).$$

How to show (rigorously but through elementary logic) that the function $$f$$ is

(1) continuous everywhere?

(2) differentiable nowhere?

This example has been given in Sec. 6.1 in the book Introduction To Real Analysis by Robert G. Bartle & Donald R. Sherbert, 4th edition. So ideally I would like to have an argument based purely on the machinary developed in the book upto this point.

However, a proof using the relevant results in the subsequent chapters and sections of the book would also be fine, provided that due references are given of all the facts used.

I do know that the infinite series in question does converge (in fact it converges absolutely). So the function is defined everywhere on the real line.

• It converges uniformly by the Weierstrass M-test, which at once implies continuity. Item (2) is the more difficult part. – Hans Lundmark Feb 8 at 20:53
• For the second question you can look at section 2 of this paper about the Weierstrass function. – aleden Feb 8 at 20:56