Nowhere continuous function

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In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers to real numbers, then f(x) is nowhere continuous if for each point x there is an ε > 0 such that for each δ > 0 we can find a point y such that |x − y| < δ and |f(x) − f(y)| ≥ ε. The import of this statement is that no matter how close we get to any fixed point, there are nearby points at which the function takes not-nearby values.

More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or the continuity definition by the definition of continuity in a topological space.

One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is written I_Q and has domain and codomain both equal to the real numbers. I_Q(x) equals 1 if x is a rational number and 0 if x is not rational. If we look at this function in the vicinity of some number y, there are two cases:

If y is rational, then f(y) = 1. To show the function is not continuous at y, we need to find an ε such that no matter how small we choose δ, there will be points z within δ of y such that f(z) is not within ε of f(y) = 1. In fact, 1/2 is such an ε. Because the irrational numbers are dense in the reals, no matter what δ we choose we can always find an irrational z within δ of y, and f(z) = 0 is at least 1/2 away from 1.
If y is irrational, then f(y) = 0. Again, we can take ε = 1/2, and this time, because the rational numbers are dense in the reals, we can pick z to be a rational number as close to y as is required. Again, f(z) = 1 is more than 1/2 away from f(y) = 0.

In plainer terms, between any two irrationals, there is a rational, and vice versa.

The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows:

$f(x)=\lim_{k\to\infty}\left(\lim_{j\to\infty}\left(\cos(k!\pi x)^{2j}\right)\right)$

(for integer k)

This shows that the Dirichlet function is a Baire class 2 function. It cannot be a Baire class 1 function because a Baire class 1 function can only be discontinuous on a meagre set.^[1]

In general, if E is any subset of a topological space X such that both E and the complement of E are dense in X, then the real-valued function which takes the value 1 on E and 0 on the complement of E will be nowhere continuous. Functions of this type were originally investigated by Dirichlet.

[edit] See also

Thomae's function (also known as the popcorn function) - a function that is continuous at all irrational numbers and discontinuous at all rational numbers.