I don't yet have a reference, but it seems that the result was first proved by Fichtenholz and Sierpiński, independently from each other. This should be mentioned in a remark to Problem 12 in:
R. Sikorski, Real Functions, Vol. 1, PWN: Warsaw, 1958 (in Polish),
at least according to the historical remark on p. 28 in:
P. Lorenc and R. Wituła, Darboux property of the nonatomic $\sigma$-additive positive and finite dimensional vector measures, Matematyka Stosowana 3 (2013), 25-36.
Unfortunately, I couldn't retrieve either a copy of Sikorski's book or simply a scan of the relevant page(s).