Some quantities related to monotonicity and bounded variation of functions (Q968838)

Some qualitative properties of real functions, like monotonicity variation can be also quantified. Quantities like modulus of decrease, increase and monotonicity for real bounded functions on a compact interval \(I\) are introduced. For example, the modulus of decrease for such function \(x\) and \(\varepsilon >0\) is the number \(d(x, \varepsilon ) = \sup \{ |x(s) -x(t) | - (x(s) - x(t)); t,s \in I, t<s, s-t < \varepsilon \}.\) Several properties of such quantities are established and relation of these quantities to the measure of noncompactness is discussed.