Computation of uniform integral means and some similar characteristics of piecewise continuous functions

We obtain formulas for the computation of a certain asymptotic characteristic of piecewise continuous functions defined on the half-line and use it to state the well-known necessary and sufficient condition on a linear homogeneous differential system for the corresponding inhomogeneous system with arbitrary inhomogeneity whose characteristic exponent is nonpositive to have a solution with nonpositive characteristic exponent. We give a new form of this necessary and sufficient condition similar to the Perron-Maizel condition for the exponential dichotomy of the system.