monotonicitás

Monotonicity is a property of a function that describes its behavior regarding increasing or decreasing values. A function is considered monotonic if it is either entirely non-increasing or entirely non-decreasing over its entire domain.

A function f is non-decreasing if for any two points x1 and x2 in its domain, whenever x1 ≤ x2, it follows that f(x1) ≤ f(x2). This means the function's values either stay the same or increase as the input increases.

Conversely, a function f is non-increasing if for any two points x1 and x2 in its domain, whenever x1 ≤ x2, it follows that f(x1) ≥ f(x2). In this case, the function's values either stay the same or decrease as the input increases.

If a function is strictly increasing (f(x1) < f(x2) whenever x1 < x2) or strictly decreasing (f(x1) > f(x2) whenever x1 < x2), it is also considered monotonic. However, the definitions of non-increasing and non-decreasing are more general as they allow for plateaus where the function's value remains constant over an interval.

The concept of monotonicity is fundamental in various areas of mathematics, including calculus, analysis, and optimization. For example, monotonic functions are guaranteed to have an inverse function if they are strictly monotonic. In calculus, the derivative of a monotonic function can provide information about its behavior; a positive derivative generally indicates a non-decreasing function, and a negative derivative suggests a non-increasing function.