Basic Concepts of Continuity

Continuity means the graph of a function has no breaks. Grasping the concept is essential for learning calculus.

Definition of Continuity

Definition of continuity at a point

Definition of continuity

Let $y = f(x)$ be defined in a neighborhood of $x_0$ . If $\lim_{x \to x_0} f(x) = f(x_0)$ , then $f(x)$ is continuous at $x_0$ .

Mathematical description

$f(x)$ is continuous at $x_0$ if and only if:

$f(x_0)$ exists (the function is defined at $x_0$ )
$\lim_{x \to x_0} f(x)$ exists
$\lim_{x \to x_0} f(x) = f(x_0)$

Epsilon–delta definition

$f(x)$ is continuous at $x_0$ iff for any $\varepsilon > 0$ there exists $\delta > 0$ such that when $|x - x_0| < \delta$ , we have $|f(x) - f(x_0)| < \varepsilon$ .

$\varepsilon$ (epsilon): Greek letter, pronounced “EP-si-lon”, often used for an arbitrarily small positive number.

$\delta$ (delta): Greek letter, pronounced “DEL-ta”, paired with $\varepsilon$ to bound the input change.

Geometric Meaning of Continuity

Intuitive view

Geometrically, continuity means the curve has no jumps or breaks at that point.

Continuous point: the curve stays connected at the point
Discontinuous point: the curve has a jump or gap

Graph features

Continuous functions: the graph is a smooth connected curve
Discontinuous functions: the graph has jumps or gaps at some points

Continuous function example

Discontinuous function example

Basic Properties of Continuity

1. Local Properties

Local boundedness: if $f$ is continuous at $x_0$ , it is bounded in a neighborhood of $x_0$
Local sign preservation: if $f$ is continuous at $x_0$ and $f(x_0) > 0$ , then $f$ stays positive in some neighborhood of $x_0$

2. Operational Properties

If $f(x)$ and $g(x)$ are continuous at $x_0$ , then:

Sum/difference: $f(x) \pm g(x)$ is continuous at $x_0$
Product: $f(x) \cdot g(x)$ is continuous at $x_0$
Quotient: $\frac{f(x)}{g(x)}$ is continuous at $x_0$ if $g(x_0) \neq 0$
Composition: $f(g(x))$ is continuous at $x_0$ if $g$ is continuous at $x_0$ and $f$ is continuous at $g(x_0)$

Exercises

Exercise 1

Determine whether $f(x) = x^2 + 1$ is continuous at $x = 2$ .

Reference Answer

Solution Approach: Check the definition of continuity.

Detailed Steps:

$f(2) = 2^2 + 1 = 5$ exists.
$\lim_{x \to 2} (x^2 + 1) = 5$ .
The limit equals the function value, so it is continuous.

Answer: Continuous.

Exercise 2

Check the continuity of $f(x) = \dfrac{1}{x}$ at $x = 0$ .

Reference Answer

Solution Approach: Check if the function is defined at the point.

Detailed Steps:

$f(0)$ is undefined.
Therefore the function is not continuous at $x = 0$ .

Answer: Discontinuous because the function is not defined at this point.

Exercise 3

For $f(x) = \begin{cases} x^2, & x \leq 1 \\ 2x - 1, & x > 1 \end{cases}$ , determine continuity at $x = 1$ .

Reference Answer

Solution Approach: Compare left limit, right limit, and function value.

Detailed Steps:

$\lim_{x \to 1^-} f(x) = 1$
$\lim_{x \to 1^+} f(x) = 1$
$f(1) = 1$

Answer: All three are equal, so continuous at $x = 1$ .

Exercise 4

For $f(x) = \begin{cases} x^2, & x \leq 0 \\ 2x + 1, & x > 0 \end{cases}$ , decide continuity at $x = 0$ .

Reference Answer (2 个标签)

continuity function continuity

Solution Approach: Compute left/right limits and the function value.

Detailed Steps:

$\lim_{x \to 0^-} f(x) = 0$
$\lim_{x \to 0^+} f(x) = 1$
$f(0) = 0$

Answer: Left and right limits are unequal, so discontinuous at $x = 0$ .

Exercise 5

Given $f(x)$ is continuous at $x = a$ and $f(a) > 0$ , prove that there exists $\delta > 0$ such that when $|x - a| < \delta$ , $f(x) > 0$ .

Reference Answer (2 个标签)

continuity function continuity

Solution Approach: Use the local sign preservation property of continuity.

Detailed Steps:

Let $\varepsilon = \dfrac{f(a)}{2} > 0$ .
By continuity, $\exists \delta > 0$ such that $|x - a| < \delta \Rightarrow |f(x) - f(a)| < \varepsilon$ .
Then $f(x) > f(a) - \varepsilon = \dfrac{f(a)}{2} > 0$ .

Answer: Such a $\delta$ exists that guarantees $f(x) > 0$ .

Exercise 6

Discuss continuity of $f(x) = \dfrac{\sin x}{x}$ at $x = 0$ .

Reference Answer (2 个标签)

continuity function continuity

Solution Approach: Check the function value and limit.

Detailed Steps:

$f(0)$ is undefined, but we can extend by setting $f(0) = 1$ .
$\lim_{x \to 0} \dfrac{\sin x}{x} = 1$ .
With $f(0)=1$ , the function becomes continuous at $0$ ; without it, it is not.

Answer: Continuous after extension with $f(0)=1$ , otherwise discontinuous.

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\varepsilon$	希腊字母	Epsilon（艾普西龙/伊普西隆）	任意小的正数
$\delta$	希腊字母	Delta（德尔塔）	与 $\varepsilon$ 搭配的正数

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
连续性	continuity	/kɒntɪˈnjuːəti/	函数在某点没有跳跃或断裂的性质
连续	continuous	/kənˈtɪnjuəs/	函数在某点满足连续性定义
连续点	continuous point	/kənˈtɪnjuəs pɔɪnt/	函数在该点连续的点
不连续点	discontinuous point	/dɪskənˈtɪnjuəs pɔɪnt/	函数在该点不连续的点
右连续	right continuous	/raɪt kənˈtɪnjuəs/	函数在某点右侧连续
左连续	left continuous	/left kənˈtɪnjuəs/	函数在某点左侧连续
局部性质	local property	/ˈləʊkəl ˈprɒpəti/	函数在某个邻域内的性质
局部有界性	local boundedness	/ˈləʊkəl ˈbaʊndɪdnəs/	函数在某个邻域内有界的性质
局部保号性	local sign preservation	/ˈləʊkəl saɪn prevəˈveɪʃən/	函数在某个邻域内保持符号的性质
运算性质	operational property	/ɒpəˈreɪʃənl ˈprɒpəti/	函数经过四则运算后仍为连续函数的性质
复合连续性	composite continuity	/ˈkɒmpəzɪt kɒntɪˈnjuːəti/	复合函数保持连续性的性质

Next连续性的判定方法

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Exploring Functions in Advanced Mathematics
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Functions are a core idea of advanced mathematics. This course walks through foundational concepts, key properties, and classic constants so you can read, reason, and compute with confidence.
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Continuity in Advanced Calculus
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A focused guide on continuity: core definitions, types of discontinuities, and continuity of elementary functions.
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