LingvoCore.com

continuous function: meaning, definition, pronunciation and examples

C2
UK/kənˈtɪn.ju.əs ˈfʌŋk.ʃən/US/kənˈtɪn.ju.əs ˈfʌŋk.ʃən/

Technical/Academic

My Flashcards

Quick answer

What does “continuous function” mean?

A mathematical function whose graph has no breaks, jumps, or holes.

Audio

Pronunciation

Definition

Meaning and Definition

A mathematical function whose graph has no breaks, jumps, or holes; a function where small changes in input produce small changes in output.

In mathematics, a function that is continuous at every point in its domain. In general usage, can describe any process or system that operates without interruption or abrupt changes.

Dialectal Variation

British vs American Usage

Differences

No significant differences in mathematical usage. In general language, British English might slightly prefer 'continuous' over 'continual' for uninterrupted processes, while American English uses both interchangeably more often.

Connotations

In both varieties, carries connotations of reliability, smoothness, and predictability in technical contexts.

Frequency

Much more frequent in academic/technical contexts than in everyday speech in both varieties.

Grammar

How to Use “continuous function” in a Sentence

be continuous at [point]remain continuous throughout [domain]ensure [something] is continuousdefine as continuous

Vocabulary

Collocations

strong
prove a continuous functiondefine a continuous functioncontinuous function on an intervaluniformly continuous function
medium
almost continuous functioncontinuous function spacecontinuous function propertycontinuous function theorem
weak
absolutely continuous functioncontinuous function analysiscontinuous function mappingcontinuous function model

Examples

Examples of “continuous function” in a Sentence

verb

British English

  • The process needs to function continuously throughout the night.
  • We must ensure the system functions continuously without interruption.

American English

  • The algorithm functions continuously once initiated.
  • The machine is designed to function continuously for 72 hours.

adverb

British English

  • The system operates continuously function-wise.
  • The data flows continuously through the function pipeline.

American English

  • The process runs continuously, functioning without pause.
  • The signal is transmitted continuously via the transfer function.

adjective

British English

  • The continuous function mapping was essential for the proof.
  • We studied continuous function spaces in the advanced course.

American English

  • The continuous function property holds for all real numbers.
  • Continuous function analysis requires careful attention to limits.

Usage

Meaning in Context

Business

Rare; might appear in data analysis contexts: 'We need continuous functions to model our sales growth.'

Academic

Very common in mathematics, physics, engineering: 'The proof requires the function to be continuous on the closed interval.'

Everyday

Very rare; if used, means 'uninterrupted process': 'The continuous function of the heating system kept the house warm.'

Technical

Extremely common with precise definition in mathematics, computer science, engineering.

Vocabulary

Synonyms of “continuous function”

Strong

uninterrupted functionseamless function

Neutral

unbroken functionsmooth functionconnected function

Weak

steady functionconsistent functionregular function

Vocabulary

Antonyms of “continuous function”

discontinuous functionbroken functioninterrupted functiondiscrete function

Watch out

Common Mistakes When Using “continuous function”

  • Using 'continuous' to mean 'constant' (a different mathematical concept).
  • Confusing 'continuous' with 'differentiable' (all differentiable functions are continuous, but not vice versa).
  • Assuming everyday 'continuous' has the same precision as mathematical 'continuous'.

FAQ

Frequently Asked Questions

No. While all differentiable functions are continuous, the converse is not true. The absolute value function |x| is continuous everywhere but not differentiable at x=0.

A function is uniformly continuous if the choice of δ in the epsilon-delta definition depends only on ε, not on the point x. All uniformly continuous functions are continuous, but not all continuous functions are uniformly continuous (e.g., f(x)=x² on ℝ).

Yes. For example, f(x)=x if x is rational, f(x)=0 if x is irrational is continuous only at x=0.

A function f is continuous at point c if for every ε>0, there exists δ>0 such that |x-c|<δ implies |f(x)-f(c)|<ε. It's continuous on its domain if it's continuous at every point.

A mathematical function whose graph has no breaks, jumps, or holes.

Continuous function: in British English it is pronounced /kənˈtɪn.ju.əs ˈfʌŋk.ʃən/, and in American English it is pronounced /kənˈtɪn.ju.əs ˈfʌŋk.ʃən/. Tap the audio buttons above to hear it.

Phrases

Idioms & Phrases

  • “As continuous as the flow of time”
  • “Continuous as a heartbeat”

Learning

Memory Aids

Mnemonic

Think of drawing a curve without lifting your pen from the paper - that's a continuous function.

Conceptual Metaphor

A PATH WITHOUT GAPS, A FLOW WITHOUT BREAKS, A SMOOTH JOURNEY

Practice

Quiz

Fill in the gap
A function has the property that lim_{x→c} f(x) = f(c) for all c in its domain.
Multiple Choice

Which of the following is NOT necessarily true for a continuous function?