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.