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 information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)