Abstract
The aim of this work is to establish the monotonicity of a Darboux, 2-injective function
Highlights
The results in this paper are related to the following classical theorem from mathematical analysis: Theorem 1
The classical Intermediate Value Theorem states that every continuous function f : I → R is a Darboux function
Function is continuous in its domain of definition. This statement is a kind of reciprocal part of the connection between the Intermediate Value Property and continuity
Summary
The results in this paper are related to the following classical theorem from mathematical analysis: Theorem 1. The classical Intermediate Value Theorem states that every continuous function f : I → R is a Darboux function. As it is expected, there exist Darboux functions that are not continuous. Function is continuous in its domain of definition This statement is a kind of reciprocal part of the connection between the Intermediate Value Property and continuity. A Darboux, injective function f : I → R is strictly monotone. We relax the injectivity condition in order to see how the monotonicity results change In this sense, we introduce the notion of a 2-injective function, that is a function that takes any value from its codomain at most twice. The example of quadratic functions are symbolic examples of 2-injective functions
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