A function f is said to be bounded on a set A if the greatest value of the function on the set is finite. This means that for any real number M > 0, there exis...
A function f is said to be bounded on a set A if the greatest value of the function on the set is finite.
This means that for any real number M > 0, there exists an element x in A such that |f(x)| > M.
In other words, the function takes on every possible value between 0 and infinity within the set A.
The boundness theorem provides an important condition for determining whether a function is continuous on a set. A function is continuous if and only if it is bounded on the set.
For example, consider the function f(x) = 1/x. This function is bounded on the interval (0, ∞) since its values can take on any real value between 0 and 1.
However, f(x) is not continuous at x = 0 because its value is undefined at that point