Defining an Antiderivative for Monotone Functions

If the author has presented no other definition of a function, then you are probably correct and it should be almost everywhere. However, it may be the case for the author's requirements it is only necessary that the function be continuous almost everywhere, in which case it is just poor writing (Off the top of my head, Vrabie has something to do with null-measure sets called a "strongly measurable function" in his book on $C_0$-semigroups; can't remember the details).

Alternatively, it may be the case that the author's definition of a function lets them get away with this statement. The construction a Cauchy sequence of "simple" functions (polynomial, piecewise-linear) to represent the functions, and taking the sequence of the antiderivatives to represent the antiderivative, with limits of sequences of functions converging in some definition of a measure on functions (sup, l-p, Lebesgue), lets one define distributional derivatives of non-differentiable things.

If someone can tell you more definitively that this is wrong, the author would probably appreciate notification of the error.