中文摘要：

下半连续函数对于最优化问题及其算法的收敛性研究具有重要意义,下半连续复合函数是其重要的组成部分。本文主要探究下半连续复合函数的闭性,证明在一定条件下半连续函数的复合保持半连续性质不变。作为对半连续函数的广义四则运算性质的补充,本文证明了下半连续函数与下半连续光滑非减函数的复合函数保持下半连续性质。同时,利用上图证明了连续函数与半连续函数、下半连续函数与下半连续且非减函数的复合保持闭性的条件。

英文摘要：

The author holds that lower semi-continuous function is of great significance for the studies of problem optimization and convergence of it＇s algorithm. Composite lower semi-continuous functions is an important part of it. This paper principally discusses the closeness of the composite function of lower semi-continuous functions. It proves that the composite of lower semi-continuous function and smooth non-decreasing lower semi-continuous function keep lower semi-continuous properties. This paper uses epigraph to show that the composite function of continuous and lower semi-continuous,lower semi-continuous and smooth non-decreasing lower semi-continuous is closed.