## Quick Reference

Let *f* be a real function and *D* a subset of its domain. If there is a point *c* in *D* such that *f*(*c*)≥*f*(*x*) for all *x* in *D*, then *f*(*c*) is the greatest value of *f* in *D*. There may be no such point: consider, for example, either the function *f* defined by *f*(*x*)=1/*x* or the function *f* defined by *f*(*x*)=*x*, with the open interval (0, 1) as *D*; or the function *f* defined by *f*(*x*)=*x*−[*x*], with the closed interval [0, 1] as *D*. If the greatest value does exist, it may be attained at more than one point of *D*.

That a continuous function on a closed interval has a greatest value is ensured by the non-elementary theorem that such a function ‘attains its bounds’. An important theorem states that a function, continuous on [*a*, *b*] and differentiable in (*a*, *b*), attains its greatest value either at a local maximum (which is a stationary point) or at an end-point of the interval.

*Subjects:*
Mathematics.

## Related content in Oxford Index

##### Reference entries

Users without a subscription are not able to see the full content. Please, subscribe or login to access all content.