Locally bounded function

In mathematics, a function is locally bounded if it is bounded around every point. A of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number.

A real-valued or complex-valued function f defined on some topological space X is called locally bounded if for any x₀ in X there exists a neighborhood A of x₀ such that f (A) is a bounded set, that is, for some number M>0 one has

for all x in A.

That is to say, for each x one can find a constant, depending on x, which is larger than all the values of the function in the neighborhood of x. Compare this with a bounded function, for which the constant does not depend on x. Obviously, if a function is bounded then it is locally bounded. The converse is not true in general.

This definition can be extended to the case when f takes values in some metric space. Then the inequality above needs to be replaced with

for all x in A, where d is the distance function in the metric space, and a is some point in the metric space. The choice of a does not affect the definition. Choosing a different a will at most increase the constant M for which this inequality is true.

is bounded, because 0≤ f (x) ≤ 1 for all x. Therefore, it is also locally bounded.

is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a, |f(x)| ≤ M in the neighborhood (a - 1,a + 1), where M = 2|a|+3.

for x ≠ 0 and taking the value 0 for x=0 is not locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.

A set (also called a family) U of real-valued or complex-valued functions defined on some topological space X is called locally bounded if for any x₀ in X there exists a neighborhood A of x₀ and a positive number M such that

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