Define a sum of non-negative numbers indexed by the uncountable set S to be

Sup (D finite subset of S) sum (d in D) r_d,

which exists if this supremum is finite.

Define a point function to be a function from [0, 1] to R that is zero everywhere except for a single point, where it takes a positive value.

Suppose we have an uncountable family of point functions f_r: [0, 1] -> R indexed by r in [0, 1]. Define the pointwise sum function S[a, b]: [0, 1] -> R as S[a, b] (x) = Sum (r in [a, b]) f_r (x).

It can be shown that if S[0, 1] is well defined, then so is S[0, a] for any a such that 0 <= a < 1.

Prove or disprove that if S[0, 1] is well defined, then for Lebesgue almost every a in [0, 1], the function S[0, a] is discontinuous at at least one point.