Summary: A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function \(f\) and the fact that the probability measure \(\lambda_{f^*}:=\lambda\circ (f^*)^{-1}\), where \(f^*(x):=(x,f(x))\) and \(\lambda\) is the Lebesgue measure, has the transport property.