Mathematical Writing - Vivaldi Franco 2014
Boundedness
Describing Functions
A set is bounded if there is an interval containing it,2 namely if
(5.6)
or
(5.7)
The two definitions are equivalent. (Think about it.) The numbers and in (5.6) are an upper and a lower bound for .
A real function is bounded if its image is a bounded set. In symbols:
For example, the sine function is bounded and the exponential is not. The periodic function displayed in Fig. 5.2 is bounded.
A function is bounded away from zero if its reciprocal is bounded. This means that for some positive constant we have for all values of . In symbols:
The hyperbolic cosine is bounded away from zero (what could be a value of in this case?) but the exponential function is not.