Some problems about the limit of a real-valued function

Dimitrije Hajduković

Kragujevac Journal of Mathematics, Tome 24 (2002) no. 1.

Résumé

Kragujevac J. Math. 24 (2002) 71-79. SOME PROBLEMS ABOUT THE LIMIT OF A REAL-VALUED FUNCTIONDimitrije HajdukovicMilana Babica 5, 51000 Banja Luka, BiH (Received June 20, 2001) 1. In [1] S. Banach solved the problem of the existence of a (non-unique) linear shift-invariant functional on the space of all bounded functions defined on the semi-axis t ł 0. 2. Let now a be sufficiently large (written a > a0 for some a0). Denote by W the real vector space of all real-valued functions on [0,Ą) and bounded on [a,Ą). This paper is organized as follows. First we will show the existence of a family of functionals on the space W containing Banach shift-invariant functionals. Next, by these functionals we shall define the limit of f(t) as t®Ą, f Î W, and show that this definition is equivalent to the classical definition of this limit. Further, we show some theorems characterizing the limit of a function f(t), t ł 0 as t®+Ą. Each of these theorems gives an answer to the question what (new) conditions must satisfy a function f Î W such that the limit of f(t) as t®+Ą exists.