A bounded function is a function whose values lie within a specific range, which means that it has an upper bound and a lower bound. Mathematically, a function f(x) is bounded if there exist real numbers M and m such that:
m ≤ f(x) ≤ M for all x in the domain of f
Bounded functions play an important role in calculus and analysis, particularly in the context of convergence and continuity. For example, the Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence, and the Extreme Value Theorem states that a continuous function on a closed interval is both bounded and attains its bounds.