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Consider the function defined by the Fourier series

$$ f(x;alpha) = sum_n=1^infty frac1n^alpha exp(i n^2 x ) , $$

where $alpha >1 $.

For what values of $alpha $ is $f$ differentiable? Based on numerics, it is conjectured that $alpha = 2$ is a critical value. For $alpha <2 $, the function is nowhere differentiable; while for $alpha >2 $, the function is differentiable almost everywhere.

The fact that $f(x,alpha)$ is everywhere non-differentiable for $alpha<2$ follows by Theorem 2.1 here but earlier versions known to Freud and Hardy suffice

Theorem [Hardy]: An integrable periodic function $f$ with Fourier series
$sum a_k sin(n_k x)$, satisfying $inflimits_kfracn_k+1n_k > 1$ is differentiable at a point only if $limlimits_krightarrowinfty a_kn_k = 0$.

For the question at the endpoint one has a mix of non-differentiability and differentiability results known, one interesting answer being the following of Gerver

Theorem [Gerver, 1970]: $f(x)=sumlimits_n=1^infty fracsinn^2 xn^2$ is differentiable at points $kpi$ if $k=frac2p+12q+1$, $p,qinmathbbZ$.