The aim of this paper is to investigate an integral mean value theorem proposed by one of the references of this paper. Unfortunately, the proof contains a gap. First, we present a counterexample which shows that this theorem fails in this form. Then, we present two improved versions of this theorem. The stability of the mean point arising from the second result concludes this paper.

The mean value theorems represent some of the most useful mathematical analysis tools. The first known result is due to Lagrange (1736–1813). In the years that followed, more mathematicians investigated this subject. As consequences of this fact, now we can find similar results, more generalizations, or extensions. Sahoo and Riedel’s book [

In [

Let

This result is very generous. Moreover, the authors obtained some interesting consequences. Unfortunately, the proof contains a gap. The aim of this paper is to present a counterexample which shows that the result from the previous theorem is not necessarily valid under this hypothesis. Afterwards, we present some conditions for which equality (

Let us consider the functions

In this section we present two valid versions of Theorem

Let

First, for every

Let

It remains to show that

As consequences of the previous theorem, we obtain the following result.

Let

We suppose that

The parents of the stability concept are considered to be the mathematicians Ulam and Hyers (see [

We want to conclude this paper with a new stability result. It is about the Hyers-Ulam stability of the mean point defined by Theorem

Let

We assume that the functions

Now, let

By using (

Now, let

The author declares that there is no conflict of interests regarding the publication of this paper.

This work has been funded by the Sectoral Operational Programme Human Resources Development 2007–2013 of the Ministry of European Funds through the Financial Agreement POSDRU 187/1.5/S/155420.