Several Glimm-type functionals for (piecewise smooth) approximate solutions
of nonlinear hyperbolic systems have been introduced in recent years.
In this paper, following a work by Baiti and Bressan on genuinely nonlinear systems
we provide a framework to prove that such functionals can be extended to general functions
with bounded variation and we investigate their lower semi-continuity properties
with respect to the strong L1topology. In particular, our result applies
to the functionals introduced by Iguchi-LeFloch and Liu-Yang for systems
with general flux-functions, as well as the functional introduced
by Baiti-LeFloch-Piccoli for nonclassical entropy solutions.
As an illustration of the use of continuous Glimm-type functionals,
we also extend a result by Bressan and Colombo for genuinely nonlinear systems,
and establish an estimate on the spreading of rarefaction waves in solutions
of hyperbolic systems with general flux-function.