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Dirichlet function)

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. 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 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 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.

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Last updated: 10-29-2005 02:13:46