In mathematics, a function is locally bounded if it is bounded around every point. A family 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. Locally bounded function A real-valued or complex-valued function defined on some topological space is called a if for any there exists a neighborhood of such that is a bounded set. That is, for some number one has In other words, for each one can find a constant, depending on which is larger than all the values of the function in the neighborhood of Compare this with a bounded function, for which the constant does not depend on 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 takes values in some metric space Then the inequality above needs to be replaced with where is some point in the metric space. The choice of does not affect the definition; choosing a different will at most increase the constant for which this inequality is true. Examples The function defined by is bounded, because for all Therefore, it is also locally bounded. The function defined by is bounded, as it becomes arbitrarily large. However, it locally bounded because for each in the neighborhood where The function defined by is neither bounded 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 be continuous where and we will show that is locally bounded at for all Taking ε = 1 in the definition of continuity, there exists such that for all with . Now by the triangle inequality, which means that is locally bounded at (taking and the neighborhood ). This argument generalizes easily to when the domain of 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 given by and for all Then is discontinuous at 0 but is locally bounded; it is locally constant apart from at zero, where we can take and the neighborhood for example. Locally bounded family A set (also called a family) U of real-valued or complex-valued