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. Therefore, no matter how close we get to any fixed point, there are even closer 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 by using the definition of continuity in a topological space.
Read more about Nowhere Continuous Function: Dirichlet Function, Hyperreal Characterisation
Famous quotes containing the words continuous and/or function:
“The habit of common and continuous speech is a symptom of mental deficiency. It proceeds from not knowing what is going on in other people’s minds.”
—Walter Bagehot (1826–1877)
“Literature does not exist in a vacuum. Writers as such have a definite social function exactly proportional to their ability as writers. This is their main use.”
—Ezra Pound (1885–1972)