A function is said to have the intermediate value property (IVP) if, for all there exists such that . In other words, has the IVP if it attains every value between and at some point in the interval .
The intermediate value theorem states that if is continuous, then has the IVP. This fact leads to the question of whether or not there exist noncontinuous functions that have the IVP. How could one exhibit such a function? This article will settle the question by constructing a collection of noncontinuous functions that have the IVP.
Let be a function. One calls strongly surjective if for any interval , the image of this interval under is . In symbols, .
Any strongly surjective function certainly satisfies the IVP on any subinterval of the real line. Thus, it suffices to construct a strongly surjective function. These functions are always, by the following argument, not continuous at any point in .
Let be a strongly surjective function. Suppose is continuous at . By the epsilon-delta formulation of continuity, this means that for any , there exists such that implies . But by strong surjectivity, This is impossible unless , which is absurd, as is a finite real number. By contradiction, we conclude is not continuous at .
In modular arithmetic, two integers and are considered to be equivalent modulo if and give the same remainder upon division by . This is the same as requiring is a multiple of . If denotes the set of multiples of , then this may be again reformulated as the condition that .
One can do something similar in a more general setting. Let be an ideal of . Declare to be equivalent if . This is an equivalence relation, and the set of equivalence classes obtained in this way is denoted .
If , then one obtains the set . This set comes equipped with a natural surjective map which sends each real number to its equivalence class in . In fact, has a property stronger than surjectivity; in fact for any interval .
Let (if is not a closed interval, replace it with a closed subinterval of itself). For any , let be a rational number such that . Then and , so any element of is mapped to by some number in .
Since the projection is surjective when restricted to any subinterval of , to construct a strongly surjective function, it would suffice to compose with a surjection . Does such a surjection exist?
Then, is a strongly surjective function. In fact, since this will hold for any bijection , the construction yields a family of strongly surjective functions, one for each such bijection.