Editing Local boundedness (section)

== Examples ==

* The family of functions <math>f_n : \R \to \R</math> <math display=block>f_n(x) = \frac{x}{n}</math> where <math>n = 1, 2, \ldots</math> is locally bounded. Indeed, if <math>x_0</math> is a real number, one can choose the neighborhood <math>A</math> to be the interval <math>\left(x_0 - a, x_0 + 1\right).</math> Then for all <math>x</math> in this interval and for all <math>n \geq 1</math> one has <math display=block>|f_n(x)| \leq M</math> with <math>M = 1 + |x_0|.</math> Moreover, the family is [[uniformly bounded]], because neither the neighborhood <math>A</math> nor the constant <math>M</math> depend on the index <math>n.</math>

* The family of functions <math>f_n : \R \to \R</math> <math display=block>f_n(x) = \frac{1}{x^2+n^2}</math> is locally bounded, if <math>n</math> is greater than zero. For any <math>x_0</math> one can choose the neighborhood <math>A</math> to be <math>\R</math> itself. Then we have <math display=block>|f_n(x)| \leq M</math> with <math>M = 1.</math> Note that the value of <math>M</math> does not depend on the choice of x<sub>0</sub> or its neighborhood <math>A.</math> This family is then not only locally bounded, it is also uniformly bounded.

* The family of functions <math>f_n : \R \to \R</math> <math display=block>f_n(x) = x+n</math> is {{em|not}} locally bounded. Indeed, for any <math>x</math> the values <math>f_n(x)</math> cannot be bounded as <math>n</math> tends toward infinity.