Editing Local boundedness (section)

==Locally bounded function==

A [[Real number|real-valued]] or [[Complex number|complex-valued]] function <math>f</math> defined on some [[topological space]] <math>X</math> is called a '''{{visible anchor|locally bounded functional}}''' if for any <math>x_0 \in X</math> there exists a [[Neighborhood (mathematics)|neighborhood]] <math>A</math> of <math>x_0</math> such that <math>f(A)</math> is a [[bounded set]]. That is, for some number <math>M > 0</math> one has
<math display=block>|f(x)| \leq M \quad \text{ for all } x \in A.</math>

In other words, for each <math>x</math> one can find a constant, depending on <math>x,</math> which is larger than all the values of the function in the neighborhood of <math>x.</math> Compare this with a [[bounded function]], for which the constant does not depend on <math>x.</math> Obviously, if a function is bounded then it is locally bounded. The converse is not true in general (see below).

This definition can be extended to the case when <math>f : X \to Y</math> takes values in some [[metric space]] <math>(Y, d).</math> Then the inequality above needs to be replaced with
<math display=block>d(f(x), y) \leq M \quad \text{ for all } x \in A,</math>
where <math>y \in Y</math> is some point in the metric space. The choice of <math>y</math> does not affect the definition; choosing a different <math>y</math> will at most increase the constant <math>r</math> for which this inequality is true.