Definition:Increasing/Real Function

Definition

Let $f$ be a real function.

Then $f$ is increasing if and only if:

$x \le y \implies \map f x \le \map f y$

Also known as

Some sources refer to an increasing (real) function as a non-decreasing (real) function, and use the term increasing real function for what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is called a strictly increasing (real) function.

Some sources give this as monotonic increasing function.

Also see

• Definition:Strictly Increasing Real Function
• Definition:Decreasing Real Function
• Definition:Monotone Real Function

• Results about increasing real functions can be found here.

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 12$: Homomorphisms: Exercise $12.8$
• 1973: G. Stephenson: Mathematical Methods for Science Students (2nd ed.) ... (previous) ... (next): Chapter $1$: Real Numbers and Functions of a Real Variable: $1.3$ Functions of a Real Variable: $\text {(j)}$ Monotonic Functions
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 12.1$
• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): Chapter $1$ Introduction: $1.7$: Terminology and Notation
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): increasing function
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): monotonic increasing function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): increasing function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): monotonic increasing function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): increasing function