Difference Between Increasing, Monotonically Increasing, and Strictly Increasing Functions: A Comprehensive Guide

In the realm of mathematics, particularly in calculus and analysis, the terms 'increasing,' 'monotonically increasing,' and 'strictly increasing' functions play a crucial role in understanding the behavior of mathematical functions. This article delves into the definitions and distinctions between these concepts, highlighting their importance, especially in competitive exams like the Joint Entrance Examination (JEE).

Increasing Functions

A function fx is said to be increasing on an interval if, for any two points x1 and x2 in that interval where x1 ≤ x2, the following condition holds:

fx1 ≤ fx2

This means that as x increases, the function value can either increase or stay constant, but it cannot decrease. This is a fundamental definition that forms the basis for understanding other related concepts.

Monotonically Increasing Functions

A function fx is monotonically increasing if it satisfies the same condition as increasing functions but with a slight distinction. It is defined as follows:

For any two points x1 and x2 in the interval where x1 ≤ x2:

fx1 ≤ fx2

It's important to note that monotonicity and strict monotonicity are often used interchangeably in many contexts.

Strictly Increasing Functions

A function fx is said to be strictly increasing on an interval if, for any two points x1 and x2 in that interval where x1 2, the following condition holds:

fx1 fx2

This means that the function actually rises as x increases and does not remain constant at any point. Strictly increasing functions are a stricter condition compared to increasing and monotonically increasing functions.

Summary

In summary, the definitions are as follows:

Increasing Function: fx1 ≤ fx2 Monotonically Increasing Function: fx1 ≤ fx2 (often used interchangeably with increasing functions) Strictly Increasing Function: fx1 fx2

Understanding these distinctions is crucial, especially in contexts like the Joint Entrance Examination (JEE), where precise definitions can significantly influence problem-solving strategies.

A Monotonically Increasing Function vs. A Strictly Increasing Function

A monotonically increasing function fx satisfies:

xy ≤ fx≥fy

A strictly increasing function fx satisfies:

xy fxy

These concepts are very similar when considering decreasing functions as well. For a monotonically decreasing function fx, the condition is:

xy ≥ fx≤fy

For a strictly decreasing function fx, the condition is:

xy > fxy

However, it's important to note that over any domain, a constant function can be said to be both monotonically increasing and decreasing. For example, if you consider a constant function c:

xy ≤ fx≥fy, where fx c and fy c xy ≥ fx≤fy, where fx c and fy c

But a constant function is never strictly increasing nor strictly decreasing on any domain, as it does not satisfy the strict inequality conditions.

Continuous vs. Discrete

In the context of continuous functions, a function fx that has f'x ≥ 0 is considered monotonically increasing. However, if f'x 0 at a single point, it means the function is not strictly increasing. For instance:

fx x3 is a strictly increasing function with domain 0 ≤ x . However, it's only a monotonically increasing function on all of [-∞, 0) and [0, ∞) because f'x 3x2, which equals 0 at x 0. fx x2 is strictly increasing on 0 ≤ x but not increasing on -∞ ≤ x as it is actually strictly decreasing there. This function also has a stationary point at x 0, just like all xn where n ≥ 0

These examples illustrate the importance of domain choice in determining the nature of a function's monotonicity.

Domain Considerations

Understanding domain considerations is crucial, especially when dealing with sequences and discrete functions. For example, a function over the natural numbers or positive integers can exhibit different behaviors compared to a continuous function over the real numbers. Consider the function an n/4. This represents a monotonically increasing sequence starting from 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2, 2, and so on.

For a function continuous on -∞ ≤ x , it cannot be both strictly increasing and strictly decreasing on the same domain without being constant for at least some intervals. This is where piecewise functions or actual constant functions over the entire domain come into play.

Functions like the floor function are not continuous over the real numbers, but they are continuous over the integers, where ε ≤ δ for any bn n/m. Any constant multiple of this is also a continuous function.

These examples highlight the subtleties in function behavior and the importance of domain choice in defining the nature of functions.