On Approximations of Semicontinuous Functions by Darboux Semicontinuous Functions

Abstract

In the whole article, the function is understood as the real valued function of real variables. It is well known that the construction of semicontinuous functions is possible either as a limit (in the sense of pointwise convergence) of the monotonous sequence of continuous functions or with assistance of the certain system of associated sets. In the first case a good understanding of the constructed function is obtained. However in the second case the Darboux property of semicontinuous function is assured when certain requirements on the system of associated sets are satisfied. Therefore the combination of both approaches seems to be the optimal method when a semicontinuous function with Darboux property is constructed. Such a combined method is used when the main theorem of this paper is proven. Later, the validity of the main theorem helps us to address the problem published by J. G. Ceder and T. L. Pearson in [2].