Thomae's function

The thoma ash function, named after the German mathematician Carl Johannes Thomae (1840-1921), is a mathematical function that is discontinuous and continuous on the irrational to the rational numbers. It is related to the Dirichlet function and how this has no practical significance, but serves as an example of continuity and other mathematical topics.

Other names on the basis of the graph are the ruler function, raindrop function, popcorn function ( popcorn in the pan ) or after John Horton Conway Stars over Babylon.

Definition

The thoma ash function as real-valued function defined by:

The thoma ash function is a simple example of a function whose set of points of discontinuity is complicated. More precisely: is continuous at all irrational numbers in [0,1] and discontinuous at all rational numbers that interval.

This can, roughly speaking, are shown as follows: If is irrational and is close to, then either irrational or a rational number with a large denominator. In both cases, is near. On the other hand a series of rational and irrational numbers to (0,1), converges towards, so that does not converge to.

Related question

Conversely, however, there is no function that is continuous on the rationals and discontinuous on the irrational numbers, because the amount of points of discontinuity is always a quantity ( set of Young), while it follows from the Baire category theorem that the set of irrational numbers no amount is.

Unstetigkeitsstellenmengen

Using a variant of the Thomistic eschen function, one can show that any subset of the actually occurs as Unstetigkeitsstellenmenge a function. Indeed, if a countable union of closed sets, so you put

By a similar argument as in the thoma eschen function you can see that the set of points of discontinuity of is.