# Bounded function

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 real number *M* such that

for all *x* in *X*.^{[1]} A function that is *not* bounded is said to be **unbounded**.^{[citation needed]}

If *f* is real-valued and *f*(*x*) ≤ *A* for all *x* in *X*, then the function is said to be **bounded (from) above** by *A*. If *f*(*x*) ≥ *B* for all *x* in *X*, then the function is said to be **bounded (from) below** by *B*. A real-valued function is bounded if and only if it is bounded from above and below.^{[1]}^{[additional citation(s) needed]}

An important special case is a **bounded sequence**, where *X* is taken to be the set **N** of natural numbers. Thus a sequence *f* = (*a*_{0}, *a*_{1}, *a*_{2}, ...) is bounded if there exists a real number *M* such that

The definition of boundedness can be generalized to functions *f : X → Y* taking values in a more general space *Y* by requiring that the image *f(X)* is a bounded set in *Y*.^{[citation needed]}

Weaker than boundedness is local boundedness. A family of bounded functions may be uniformly bounded.

A bounded operator *T : X → Y* is not a bounded function in the sense of this page's definition (unless *T = 0*), but has the weaker property of **preserving boundedness**: Bounded sets *M ⊆ X* are mapped to bounded sets *T(M) ⊆ Y.* This definition can be extended to any function *f* : *X* → *Y* if *X* and *Y* allow for the concept of a bounded set. Boundedness can also be determined by looking at a graph.^{[citation needed]}