Bounded function

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In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a number M>0 such that

|f(x)|\le M

for all x in X.

Sometimes, if f(x)\le A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x)\ge 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.

An important special case is a bounded sequence, where X is taken to be the set N of natural numbers. Thus a sequence f = ( a0, a1, a2, ... ) is bounded if there exists a number M > 0 such that

|an| ≤ M

for every natural number n. The set of all bounded sequences, equipped with a vector space structure, forms a sequence space.

This definition can be extended to functions taking values in a metric space Y. Then the inequality above is replaced with

d(f(x), a)\le M

for some a in Y, M>0, and for all x in X.

[edit] Examples

  • 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
f(x)=\frac{1}{x^2-1}

defined for all real x which do not equal −1 or 1 is not bounded. 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, ∞).

  • The function
f(x)=\frac{1}{x^2+1}

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 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.
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