Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B. The concept should not be confused with that of a bounded operator. for all x in X. If this is the case, there is also such an M for each other a, by the t...