Nowhere continuous 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 x there is some 0"}}'>ε>0 such that for every 0,"}}'>δ>0, we can find a point |f(x)−f(y)|≥ε. Therefore, no matter how close it gets 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.

𝟏ℚ(x) is equal to 0 otherwise.

More generally, if E and the complement of 1 on E will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet.

f:ℝ→ℝ is called an x↦cx, where c:=L(1)).

Although every linear map is additive, not all additive maps are linear. An additive map f:ℝ→ℝ to any real scalar multiple of the rational numbers f|rℚ:rℚ→ℝ to the set x∈ℝ, x is also contained in some dense subset x≠0, and take x is infinitely close to a  –even if a real function ℝ such that the restriction of

1. ↑ 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.