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for all x in X. A function that is not bounded is said to be unbounded.
Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B.
The concept should not be confused with that of a bounded operator.
This definition can be extended to functions taking values in a metric space Y. Such a function f defined on some set X is called bounded if for some a in Y there exists a real number M such that its distance function d ("distance") is less than M, i.e.
for all x in X.
If this is the case, there is also such an M for each other a, by the triangle inequality.
- The function f : R → R defined by f(x) = sin(x) 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, [2, ∞) or (−∞, −2].
- The function
- defined for all real x is bounded.
- Every continuous function f : [0, 1] → 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 (cf. Dirichlet function) is bounded. Thus, a function does not need to be "nice" in order to be bounded. The set of all bounded functions defined on [0, 1] is much bigger than the set of continuous functions on that interval.