Local Boundedness - Locally Bounded Function

Locally Bounded Function

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 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.

Read more about this topic:  Local Boundedness

Famous quotes containing the words locally, bounded and/or function:

    To see ourselves as others see us can be eye-opening. To see others as sharing a nature with ourselves is the merest decency. But it is from the far more difficult achievement of seeing ourselves amongst others, as a local example of the forms human life has locally taken, a case among cases, a world among worlds, that the largeness of mind, without which objectivity is self- congratulation and tolerance a sham, comes.
    Clifford Geertz (b. 1926)

    I could be bounded in a nutshell and count myself a king of
    infinite space, were it not that I have bad dreams.
    William Shakespeare (1564–1616)

    The press and politicians. A delicate relationship. Too close, and danger ensues. Too far apart and democracy itself cannot function without the essential exchange of information. Creative leaks, a discreet lunch, interchange in the Lobby, the art of the unattributable telephone call, late at night.
    Howard Brenton (b. 1942)