In Weierstrass function defined as $$f(x)=\sum _{n=0}^{\infty }a^{n}\cos(b^{n}\pi x),$$ we have the requirement that where $0<a<1$, $b$ is a positive odd integer, and $ab>1+{\frac {3}{2}}\pi$.
But what will happen if $b$ is an even integer? I know from the proof that the oddness is used to reduce $(-1)^b$ to $-1$. However, I plotted the function when $b=8$, it seems still pathological (continuous&indifferentiable everywhere). I guess it has to do with asymptotic behavior.