Editing Local boundedness (section)

== Examples ==

* The function <math>f : \R \to \R</math> defined by <math display=block>f(x) = \frac{1}{x^2+1}</math> is bounded, because <math>0 \leq f(x) \leq 1</math> for all <math>x.</math> Therefore, it is also locally bounded.

* The function <math>f : \R \to \R</math> defined by <math display=block>f(x) = 2x+3</math> is {{em|not}} bounded, as it becomes arbitrarily large. However, it {{em|is}} locally bounded because for each <math>a,</math> <math>|f(x)| \leq M</math> in the neighborhood <math>(a - 1, a + 1),</math> where <math>M = 2|a| + 5.</math>

* The function <math>f : \R \to \R</math> defined by <math display=block>f(x) = \begin{cases}
  \frac{1}{x}, & \mbox{if } x \neq 0, \\
  0,  & \mbox{if } x = 0 
\end{cases}
</math> is neither bounded {{em|nor}} locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.

* Any continuous function is locally bounded. Here is a proof for functions of a real variable. Let <math>f : U \to \R</math> be continuous where <math>U \subseteq \R,</math> and we will show that <math>f</math> is locally bounded at <math>a</math> for all <math>a \in U</math> Taking ε = 1 in the definition of continuity, there exists <math>\delta > 0</math> such that <math>|f(x) - f(a)| < 1</math> for all <math>x \in U</math> with <math>|x - a| < \delta</math>. Now by the [[triangle inequality]], <math>|f(x)| = |f(x) - f(a) + f(a)| \leq |f(x) - f(a)| + |f(a)| < 1 + |f(a)|,</math> which means that <math>f</math> is locally bounded at <math>a</math> (taking <math>M = 1 + |f(a)|</math> and the neighborhood <math>(a - \delta, a + \delta)</math>). This argument generalizes easily to when the domain of <math>f</math> is any topological space.

* The converse of the above result is not true however; that is, a discontinuous function may be locally bounded. For example consider the function <math>f : \R \to \R</math> given by <math>f(0) = 1</math> and <math>f(x) = 0</math> for all <math>x \neq 0.</math> Then <math>f</math> is discontinuous at 0 but <math>f</math> is locally bounded; it is locally constant apart from at zero, where we can take <math>M = 1</math> and the neighborhood <math>(-1, 1),</math> for example.