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Nowhere continuous
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Nowhere continuous

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 |xy|<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 IQ and has domain and range both equal to the real numbers. IQ(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:

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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