In this paper, we propose new semiparametric procedures for making inference on linear functionals and their functions of two semicontinuous populations. The distribution of each population is usually characterized by a mixture of a discrete point mass at zero and a continuous skewed positive component, and hence such distribution is semicontinuous in the nature. To utilize the information from both populations, we model the positive components of the two mixture distributions via a semiparametric density ratio model. Under this model setup, we construct the maximum empirical likelihood estimators of the linear functionals and their functions, and establish the asymptotic normality of the proposed estimators. We show the proposed estimators of the linear functionals are more efficient than the fully nonparametric ones. The developed asymptotic results enable us to construct confidence regions and perform hypothesis tests for the linear functionals and their functions. We further apply these results to several important summary quantities such as the moments, the mean ratio, the coefficient of variation, and the generalized entropy class of inequality measures. Simulation studies demonstrate the advantages of our proposed semiparametric method over some existing methods. Two real data examples are provided for illustration.