A bounded function is a real-valued function or complex function 
on a domain 
for which there is a finite constant 
such that
 |
for every 
. Here 
denotes absolute value
in the real case and complex modulus in the complex
case. Equivalently, the range of 
is a bounded set.
A real-valued function 
is bounded above if there is a real upper
bound 
with 
for every 
. It is bounded below if there is a real lower
bound 
with 
for every 
. For such a function, boundedness is equivalent
to having both bounds.
Boundedness depends on the domain. For example, 
is bounded on 
. On 
, it is bounded below by 1 and has no upper
bound. A real-valued function that is
continuous on a compact
set is bounded by the extreme value theorem,
but a bounded function need not be continuous.
