Bounded Functions

A bounded function is a mathematical function whose outputs stay within certain fixed limits; they do not shoot off to infinity or drop indefinitely. Think of it like a thermostat-controlled heater that maintains room temperature within a specific range—it never gets too hot or too cold. In formal terms, there are real numbers, called bounds, that the function's values do not exceed. This concept helps in analyzing functions by ensuring their values remain manageable and predictable, which is essential in many areas of mathematics and applied sciences.