We give characterizations of the differentiability points and the non-differentiability points of the Riesz-N$'{a}$gy-Tak$'{a}$cs(RNT) singulr function using the distribution sets in the unit interval. Using characterizations, we show that the Hausdorff dimension of the non-differentiability points of the RNT singular function is greater than 0 and the packing dimension of the infinite derivative points of the RNT singular function is less than 1. Further the RNT singular function is nowhere differentiable in the sense of topological magnitude, which leads to that the packing dimension of the non-differentiability points of the RNT singular function is 1. Finally we show that our characterizations generalize a recent result from the ($ au$, $ au$ - 1)-expansion associated with the RNT singular function adding a new result for a sufficient condition for the non-differentiability points.