  For any transitive piecewise monotonic map for which the set of periodic
measures is dense in the set of ergodic invariant measures (such as monotonic
mod one transformations and piecewise monotonic maps with two monotonic
pieces), we show that the set of points for which the Birkhoff average of a
continuous function does not exist (called the irregular set) is either empty
or has full topological entropy. This generalizes Thompson's theorem for
irregular sets of $\beta$-transformations, and reduces a complete description
of irregular sets of transitive piecewise monotonic maps to Hofbauer-Raith
problem on the density of periodic measures.
