Examples
- The function defined by is bounded. The sine function is no longer bounded if it is defined over the set of all complex numbers.
- The function
defined for all real x except for −1 and 1 is unbounded. As x gets closer to −1 or to 1, the values of this function get larger and larger in magnitude. This function can be made bounded if one considers its domain to be, for example, .
- The function
defined for all real x is bounded.
- Every continuous function f: → R is bounded. This is really a special case of a more general fact: Every continuous function from a compact space into a metric space is bounded.
- The function f which takes the value 0 for x rational number and 1 for x irrational number is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on is much bigger than the set of continuous functions on that interval.
Read more about this topic: Bounded Function
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