The Cramer transform introduced in large deviations theory sends classical probabilities (resp. finite positive measures) into (min,+) probabilities (resp. finite measures) also called cost measures. We study its continuity when the two spaces of measures are endowed with the weak convergence topology. We prove that the Cramer transform is continuous in the subspace of logconcave measures and show counter examples in the opposite case. Moreover, in finite dimension, the Cramer transform is bicontinuous. Then, logconcave measures may be identified with lower semicontinuous convex functions.