An interval map with only one discontinuity is isomorphicto a rotation of the circle, and has continuouseigenfunctions. What we show here is that for almost every choiceof lengths of the intervals, this is the only way an irreducible intervalexchangecan have a somewhere continuous eigenfunction. We show slightly more,considering certain towers over the interval exchange, showingthat outside of a set of choices for interval lengths of measure zerothese have asomewhere continuous eigenfunction only if they are isomorphic to either arotation,or a tower of constant height over an interval exchange.