"Almost everywhere" means "everywhere except on a set of measure 0", in the Lebesgue measure sense.

Here's an example of a Riemann integrable function w/ infinitely many discontinuities: https://en.wikipedia.org/wiki/Thomae%27s_function

Anyone interested in this should check out the Prologue to Lebesgue's 1901 paper: http://scratchpost.dreamhosters.com/math/Lebesgue_Integral.p...

It gives several reasons why we "knew" the Riemann integral wasn't capturing the full notion of integral / antiderivative