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.