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.
Boundedness depends on the domain. For example, 
is bounded on 
but not on 
. 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.
