What is the difference between the traditional and modern definitions of a function?

The traditional definition of a function as a change in motion does not fully reveal the nature of the concept of function. Recent mathematical developments have defined a function as a kind of mapping between two sets, according to which a function is a particular kind of mapping between two sets of numbers (or some subset thereof), which has led to a deeper understanding of the concept of function.

For example, for the function f(x) = 1 (when x is a rational number)

= 0 (when x is an irrational number)

If one looks at it in terms of changes in motion, it is poorly interpreted and seems far-fetched. But if you look at it in terms of mapping, it seems very natural.

The starting point of the traditional definition is the point of view of the change of motion, there is an obvious physical meaning, which corresponds the two variables x, y which are interdependent in the process of change of motion; the recent definition makes use of the point of view of the set, the correspondence, the law of correspondence is the correspondence of any element in the original set of the image with the only certain element in the set of the image, shedding the physical meaning and rising to the purely mathematical Perspective