Why If x is rational set fx = 0; if x is irrational set fx = 1

The given function f(x) is defined in such a way that it assigns a value of 0 to rational numbers and 1 to irrational numbers. This function can be used to distinguish between rational and irrational numbers.

The reason for setting f(x) = 0 for rational numbers is that rational numbers can be expressed as a ratio of two integers, and they have a finite or repeating decimal representation. By assigning a value of 0 to rational numbers, the function differentiates them from irrational numbers.

On the other hand, irrational numbers cannot be expressed as a ratio of two integers, and they have an infinite and non-repeating decimal representation. By setting f(x) = 1 for irrational numbers, the function distinguishes them from rational numbers.

This function can be useful in various mathematical contexts where it is necessary to categorize numbers based on their rationality or irrationality