  In this paper, we prove a criterion for existence of continuous non constant
eigenfunctions for interval exchange transformations, that is for non
topologically weak mixing. We first construct, for any m>3, uniquely ergodic
interval exchange transformations of rank 2 with irrational eigenvalues
associated to continuous eigenfunctions, so that are not topologically weak
mixing, this answers to a question of Ferenczi and Zamboni. Then, we construct,
for any even number m>3, interval exchange transformations of rank 2 with both
irrational eigenvalues (associated to continuous eigenfunctions) and non
trivial rational eigenvalues (associated to piecewise continuous
eigenfunctions). Moreover these examples can be chosen to be either uniquely
ergodic or non minimal.
