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)
“Me, what’s that after all? An arbitrary limitation of being bounded by the people before and after and on either side. Where they leave off, I begin, and vice versa.”
—Russell Hoban (b. 1925)
“We are thus able to distinguish thinking as the function which is to a large extent linguistic.”
—Benjamin Lee Whorf (1897–1934)