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Pathological cases — Dirichlet function
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The Dirichlet function has the strange property that it is nowhere continuous. The function is defined for every possible value of x, but the function is not continuous anywhere on the number line. It is a piecewise function defined by:
The stylised letter Q represents the set of rational numbers, so this function takes the value 1 if x is rational, and 0 if x is irrational. It is sometimes called the indicator function of the set of rational numbers, written as:
Despite its appearance, this is a function!
Properties of the real number line
Every real number is either rational or irrational. A rational number can be expressed as the ratio of two integers, for example 1/3 or 17/100. An irrational number cannot be expressed as the ratio of two integers. Well-known examples of irrational numbers are the square root of 2 and the mathematical constant π. But of course, there are infinitely many irrational numbers that don’t have special names.
The real numbers are often represented by a line. This diagram shows a section of the real number line with a few values marked on it: