Exercise 5.4.8

The critical part to showing that $g(x)$ is not differentiable at nondyadic points was showing that

does not converge to zero, preventing the limit defining the derivative to exist. For the case ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=0}^{\infty }(1∕2^n)h_n(3^{n}x)$, the above term would diverge to infinity, since $y_n-x_n$ would decrease by a factor of 3 on each iteration while $h_n(y_n)-h_n(x_n)$ would only decrease by a factor of 2. For similar reasons, in the case of ${\sum <!--\; FUNCTION\; APPLICATION\; -->}_{n=0}^{\infty }(1∕3^n)h_n(2^{n}x)$, the above term would converge to 0, and the argument is no longer valid.