Monotonic function

Monotonic function

A monotonic function is a function between ordered sets that preserves or reverses the given order. This concept is one of the central ideas in mathematical analysis and is particularly important in the theory of calculus and real analysis. In simple terms, a function is monotonic if its graph does not change direction but either goes up or goes down or remains constant as it moves along the x-axis.

Definition

Formally, a function f : A → B is called monotonic (or monotone) if, for all x and y in the domain A, one of the following conditions holds:

• Monotonically increasing: If x ≤ y, then f(x) ≤ f(y). If the function never decreases, it is called strictly increasing if f(x) < f(y) whenever x < y.
• Monotonically decreasing: If x ≤ y, then f(x) ≥ f(y). It is called strictly decreasing if f(x) > f(y) whenever x < y.

Types of Monotonic Functions

Monotonic functions can be broadly classified into two categories:

• Increasing Monotonic Function: These functions have a non-decreasing value as their input increases. They are further divided into strictly increasing and non-strictly (or weakly) increasing functions.
• Decreasing Monotonic Function: These functions have a non-increasing value as their input increases. Similar to increasing functions, they can be strictly decreasing or non-strictly decreasing.

Properties

Monotonic functions have several important properties that make them useful in analysis:

• If a function is monotonic, then it is injective (one-to-one) if and only if it is strictly monotonic.
• Monotonic functions on an interval are bounded on that interval and have limits at every point and at infinity.
• The inverse of a strictly monotonic function is also strictly monotonic.

Applications

Monotonic functions are utilized in various fields of mathematics and its applications:

• In Calculus, they are used to determine the convergence of sequences and series.
• In optimization, monotonic functions are important in establishing the existence of optimal solutions to problems.
• In Economics, they are used to model consumer preferences, where the monotonicity assumption implies that more is always preferred to less.

Examples

• The function f(x) = x^2 is strictly increasing on the interval [0, ∞) and strictly decreasing on the interval (-∞, 0].
• The natural logarithm function ln(x) is strictly increasing on its domain (0, ∞).

See Also

• Function
• Continuous function
• Injective function
• Surjective function
• Bijective function