Examples
- The function f: R → R defined by
is bounded, because 0≤ f (x) ≤ 1 for all x. Therefore it is also locally bounded.
- The function f: R → R defined by
is not bounded, as it becomes arbitrarily large. However, it is locally bounded because for each a, |f(x)| ≤ M in the neighborhood (a - 1,a + 1), where M = 2|a|+5.
- The function f:R → R defined by
for x ≠ 0 and taking the value 0 for x=0 is not locally bounded. In any neighborhood of 0 this function takes values of arbitrarily large magnitude.
Read more about this topic: Local Boundedness
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