A linear function f: X → Y is continuous if and only if it is continuous at 0.
A linear function f: X → Y is bounded if there exists a constant M such that
|| f(x) || ≤ M ||x|| for every x ∊ X
Note that boundedness does not imply that the range f (X) is bounded but rather that f (S) is bounded for every bounded set S. As the following exercise demonstrates, boundedness is equivalent to continuity for linear functions. Consequently these two terms are used interchangeably in practice.