Basic Concepts of Continuity

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

连续性的定义

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:

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

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 $|f(x) - f(x_0)| < \varepsilon$.

连续性的几何意义

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

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

Continuous function example

Discontinuous function example

连续性的基本性质

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:

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

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练习题

练习 1

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

参考答案

思路：Check definition of continuity.

步骤：

1. $f(2) = 2^2 + 1 = 5$ exists.
2. $\lim\_{x \to 2} (x^2 + 1) = 5$.
3. Limit equals the function value, so it is continuous.

答案：连续。

练习 2

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

参考答案

思路：See if the function is defined at the point.

步骤：

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

答案：不连续，因为函数在该点无定义。

练习 3

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

参考答案

思路：Compare left limit, right limit, and function value.

步骤：

1. $\lim\_{x \to 1^-} f(x) = 1$
2. $\lim\_{x \to 1^+} f(x) = 1$
3. $f(1) = 1$

答案：三者相等，故在 $x = 1$ 处连续。

考研真题

真题 1

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

参考答案

思路：Compute left/right limits and the function value.

步骤：

1. $\lim\_{x \to 0^-} f(x) = 0$
2. $\lim\_{x \to 0^+} f(x) = 1$
3. $f(0) = 0$

答案：左右极限不等，故 $x = 0$ 处不连续。

真题 2

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

参考答案

思路：Use local sign preservation of continuity.

步骤：

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

答案：存在这样的 $\delta$，保证 $f(x) > 0$。

真题 3

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

参考答案

思路：Check definition value and limit.

步骤：

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

答案：延拓为 $f(0)=1$ 后连续，否则不连续。

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\varepsilon$	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

中英对照

中文术语	英文术语	音标	说明
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连续函数的例子