A variant of the two-dimensional Riemann integral

For a va riant of the two-d imensional Ri e mann int egra l suggested by S. Marcus , it is shown that the only integrab le fun c tions which are continuous (o r merely continuou s se parately in one of th e variables) are the cons ta nt fun ctions . The int egrab le di scontinuou s functions a re proven to be cons ta nt except poss ib ly on a se t which is "s ma ll" in a sense made precise in the paper.