The stability of some points arising from continuous, differential and integral expressions

Abstract

The aim of this paper is to establish some new stability results on the intermediary points of Darboux functions and continuous functions, on the extreme points, the stationary points and the inflection points, on the intermediary points arising from several versions of the Mean Value Theorem and from some Riemann integrals. Our original results are presented as fourteen theorems. Some of them are new, while the others are improvements of classical results stated by Hyers (J Math Anal Appl 36:622–626, 1971), Hyers (J Math Anal Appl 62:530–537, 1978), but also of recent results due to Das et al. (Appl Math Lett 16:269–271, 2003), Găvruţă et al. (Ann Funct Anal 1:68–74 2010) and Peter and Popa (Publ Math Debr 83:1–10, 2013). More precisely, our improvements consist in weakening the conditions on the involved functions with at least one degree of smoothness. For sake of clearness, the proofs of our results are presented in a separate section at the end of this work.