A Semi-Discrete Analog to The Mean Value Theorem
The mean value theorem states that if a function is differentiable in an open interval and continuous in its closure , then there exists a point such that . This theorem's semi-discrete analog suggests that if a function is tendable in , continuous in , and also that , then for each there exists a point , such that , where is the function's tendency, and "tendability of a function" means that the tendency operator can be applied to the function. The definition of the tendency operator is given in the Demonstration "Detachment and Tendency of a Single Variable Function". A comparison between the mean value theorem and its semi-discrete analog is found in the details. When compared to the original theorem, this version depicts a trade-off between the functions for which the theorem's correctness holds and the kind of information that the theorem gives. This version is called semi-discrete due to the decisive role of the function, whose image is finite. In this Demonstration, you can vary the parameters , , and from the theorem's statement to better understand the theorem. The highlighted yellow points are those whose existence is assured by the theorem.
Snapshot 1: setting , , yields only one highlighted point; here the one-sided detachments are omitted
Snapshot 2: setting , , yields two highlighted points; here the one-sided detachments, from which the tendency is calculated, are shown
Snapshot 3: setting , , yields one highlighted point; here the one-sided detachments, from which the tendency is calculated, are shown; although there are two points in the chosen interval whose image is , only one of them is highlighted—the tendency at the other point is zero
A short and to the point comparison between the mean value theorem and its semi-discrete analog is depicted in this link.