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
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