According to the definite condition, the paper USES the theorem of supremum and infimum to study characteristic of mapping function and get its important law of the mapping.

运用上确界与下确界存在定理，在一定条件下，研究了连续函数的映射特点，得到一个连续函数所独有的映射规律。

The infimum and supremum of operators, and generalized inverse of operators are heated topics in operator theory and also have important value in both theory and application.

算子的确界性质和广义逆是近年来算子理论中比较活跃的一些研究课题，在算子理论的研究中有着重要的理论价值和应用价值。