In Hengartner, W. and Theodorescu, R. (1973). Concentration Functions. Academic Press, New York. MR033144
they study the continuity properties and when the concentration function is zero even for the multivariate case in THEOREM 1.7.4..
The main idea goes through showing that the maximum is attained for each $t>0$ and so as $t\to 0$, the concentration function goes to zero.
The article "Maximal Inequalities and Some Applications" has many other references too.