The comment of Aakumadula is enough to see that is not continuous for $ \alpha < 1/2 $. By expressing $ \sin (nx) $ in terms of complex exponentials by Euler's formula we notice that this can be expressed as a difference between two periodic zeta-functions (special cases of the Lerch zeta-function), and by applying the functional equation of the periodic zeta-function (or Lerch) this can be expressed as a sum $a(\alpha) \zeta(1-\alpha,1-x/(2 \pi))+ b(\alpha) \zeta(1-\alpha,x/(2 \pi) ) $ where $\zeta(s,y)=\sum_{n=1}^\infty (n+y)^{-s}$ denote the Hurwitz zeta-function and $a(\alpha),b(\alpha)$ are some gamma-factors. The discontinuity can now be seen from the fact that $\zeta( 1 - \alpha,y)=y^{\alpha-1}+O(1)$ for $ 0 < y < 1 $ and any fixed $ 0 < \alpha < 1 $.
The choice of which functional equation we want to use is somewhat arbitrary (it all amounts down to the Poisson summation formula), for example we can also use the functional equation for the Hurwitz zeta-function, see e.g eq 8 in the link below, and the fact that cos is an even function and sin is an odd function.