Continuity — Mathematics Knowledge System

Overview

A function is continuous at a point if you can draw its graph through that point without lifting your pen — no jumps, holes, or asymptotes. Formally, continuity means the limit of the function as x approaches a point equals the function's actual value at that point.

Intuition

A continuous function has no surprises: as you slide the input a tiny bit, the output only slides a tiny bit too — never jumps. Think of temperature over the course of a day: it changes smoothly, never teleporting from 60°F to 90°F between two consecutive seconds. Contrast that with a step function like a parking garage's price schedule, which jumps discontinuously at each hour boundary.

Interactive Graph

Formal Definition

Definition

f is continuous at a if all three conditions hold:

\text{1. } f(a) \text{ is defined}

The function has a value at the point

\text{2. } \lim_{x \to a} f(x) \text{ exists}

The limit exists at that point

\text{3. } \lim_{x \to a} f(x) = f(a)

The limit equals the actual function value

Notation

Notation	Meaning
$f \in C(a)$	f is continuous at a
$C^0([a,b])$	The set of functions continuous on the closed interval [a,b]

Properties

Sum/product/quotient of continuous functions

If f and g are continuous at a, so are f+g, f-g, f \cdot g, and f/g (where g(a)\neq 0).

Composition

If g is continuous at a and f is continuous at g(a), then f \circ g is continuous at a.

Polynomials

Every polynomial is continuous everywhere on ℝ.

Theorems

Theorem 1: Intermediate Value Theorem

If f is continuous on [a,b] and N is any value between f(a) and f(b), then there exists c \in [a,b] with f(c) = N.

Theorem 2: Extreme Value Theorem

If f is continuous on a closed interval [a,b], then f attains both a maximum and a minimum value on that interval.

Applications

Control systems require continuity guarantees to ensure predictable, non-jumping physical responses.

Worked Examples

f(0) is undefined — division by zero — so condition 1 fails immediately.
$f(0) \text{ is undefined}$

Answer: No. f is not continuous at x=0 (it isn't even defined there).

Practice Problems

Difficulty 3/10

The floor function f(x) = ⌊x⌋ is discontinuous at:

No points — it's continuous everywhereEvery real numberEvery integerEvery irrational number

Difficulty 5/10

An electricity tariff charges $0.20/kWh for the first 100 kWh and $0.30/kWh beyond that. Is the total-cost function continuous at 100 kWh? Is it differentiable there?

Difficulty 6/10

A continuous temperature reading rises from −2°C at 6 am to 5°C at noon. The Intermediate Value Theorem guarantees what, and why does continuity matter?

Common Mistakes

Common Mistake

Assuming a function is continuous just because it 'looks smooth' on a graphing calculator's zoomed-out view.

Removable discontinuities (holes) can be invisible at low zoom. Always check the formal definition at suspicious points like denominators equal to zero.

Quiz

Which of these is NOT one of the three conditions for continuity at a?

f(a) is definedf is differentiable at alim(x→a) f(x) existslim(x→a) f(x) = f(a)

Flashcards

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Historical Background

Euler used an informal notion of continuity (a curve drawn by 'free motion of the hand') in the 18th century. The rigorous definition tying continuity directly to limits came from Bolzano (1817) and independently Cauchy (1821), and was sharpened into the modern epsilon-delta form by Weierstrass alongside his work on limits.

1817
Bolzano gives an early rigorous definition and proves the intermediate value theorem
Bernard Bolzano
1821
Cauchy formalizes continuity using limits in Cours d'Analyse
Augustin-Louis Cauchy

Summary

f is continuous at a when f(a) is defined, the limit exists there, and the limit equals f(a).
Continuity is necessary (but not sufficient) for differentiability.
Sums, products, quotients, and compositions of continuous functions are continuous.
Intermediate Value Theorem: a continuous function hits every value between two endpoints.
All polynomials are continuous everywhere on ℝ.

References

WebsiteWikipedia — Continuous function