strictly increasing function: meaning, definition, pronunciation and examples
C1+Formal, Academic, Technical
Quick answer
What does “strictly increasing function” mean?
A mathematical function where, for any two points in its domain, if x1 < x2 then f(x1) < f(x2). The output always increases as the input increases.
Audio
Pronunciation
Definition
Meaning and Definition
A mathematical function where, for any two points in its domain, if x1 < x2 then f(x1) < f(x2). The output always increases as the input increases.
A fundamental concept in calculus and analysis describing a monotonic relationship where the function's value never stays the same or decreases. It is a key property for defining invertibility and understanding rates of change.
Dialectal Variation
British vs American Usage
Differences
No significant lexical or conceptual differences. Spelling of related words follows regional norms (e.g., BrE 'monotonic', AmE also 'monotonic').
Connotations
Identical technical meaning and formality.
Frequency
Equal frequency within mathematical contexts in both varieties.
Grammar
How to Use “strictly increasing function” in a Sentence
[The/This/That] function is strictly increasing on [interval/domain].A strictly increasing function [has property/implies result].Vocabulary
Collocations
Examples
Examples of “strictly increasing function” in a Sentence
verb
British English
- The sequence is proved to strictly increase.
American English
- The function strictly increases across the domain.
adverb
British English
- The values change strictly increasingly.
American English
- The curve moves strictly increasingly to the right.
adjective
British English
- We need a strictly increasing utility function.
American English
- The model assumes a strictly increasing relationship.
Usage
Meaning in Context
Business
Virtually never used.
Academic
Core terminology in mathematics, economics (utility functions), and engineering.
Everyday
Not used.
Technical
Primary context. Used in proofs, algorithm analysis (complexity), and modelling.
Vocabulary
Synonyms of “strictly increasing function”
Strong
Neutral
Weak
Vocabulary
Antonyms of “strictly increasing function”
Watch out
Common Mistakes When Using “strictly increasing function”
- Using 'strictly increasing' to describe a function that plateaus (e.g., a step function).
- Confusing it with 'injective' or 'one-to-one' (all strictly increasing functions are injective, but not all injective functions are strictly increasing).
- Omitting 'strictly' when the strict inequality is intended.
FAQ
Frequently Asked Questions
In precise mathematical English, 'increasing' can be ambiguous. 'Strictly increasing' explicitly means f(x1) < f(x2) for x1 < x2. Sometimes 'increasing' alone means 'non-decreasing' (≤), so 'strictly' is used for clarity.
Yes, but only at isolated points. For example, f(x) = x^3 is strictly increasing on all real numbers, but its derivative at x=0 is 0.
'Monotonic increasing' is often used synonymously with 'non-decreasing' (allowing equality). 'Strictly increasing' is a stronger condition that excludes constant sections.
It guarantees the function is one-to-one (injective), and therefore has an inverse function on its range. It is also crucial in optimization, probability (quantile functions), and defining order isomorphisms.
A mathematical function where, for any two points in its domain, if x1 < x2 then f(x1) < f(x2). The output always increases as the input increases.
Strictly increasing function is usually formal, academic, technical in register.
Strictly increasing function: in British English it is pronounced /ˈstrɪktli ɪnˈkriːsɪŋ ˈfʌŋkʃən/, and in American English it is pronounced /ˈstrɪktli ɪnˈkrisɪŋ ˈfʌŋkʃən/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of a hill you can only walk UP, never down or flat. If you take a step forward (increase x), you must also step higher (increase f(x)).
Conceptual Metaphor
A ONE-WAY UPWARD LADDER: each step is higher than the last, with no option to step down or stay on the same rung.
Practice
Quiz
Which of the following best describes a 'strictly increasing function'?