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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 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 0 < |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.
Dirichlet function
One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as IQ or 1Q and has ___domain and codomain both equal to the real numbers. IQ(x) equals 1 if x is a rational number and 0 if x is not rational.
More generally, 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 Peter Gustav Lejeune Dirichlet.[1]
Hyperreal characterisation
A real function f is nowhere continuous if its natural hyperreal extension has the property that every x is infinitely close to a y such that the difference f(x) − f(y) is appreciable (i.e., not infinitesimal).
See also
- Blumberg theorem – even if a real function f : ℝ → ℝ is nowhere continuous, there is a dense subset D of ℝ such that the restriction of f to D is continuous.
- Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.
- Weierstrass function – a function continuous everywhere (inside its ___domain) and differentiable nowhere.
References
- ^ Lejeune Dirichlet, Peter Gustav (1829). "Sur la convergence des séries trigonométriques qui servent à représenter une fonction arbitraire entre des limites données". Journal für die reine und angewandte Mathematik. 4: 157–169.