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.

  • 1
    $\begingroup$ It converges uniformly by the Weierstrass M-test, which at once implies continuity. Item (2) is the more difficult part. $\endgroup$ – Hans Lundmark Feb 8 at 20:53
  • 5
    $\begingroup$ For the second question you can look at section 2 of this paper about the Weierstrass function. $\endgroup$ – aleden Feb 8 at 20:56

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