Exploring Types of Discontinuities · 探索不连续性的类型
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| discontinuity/dɪskɒntɪˈnjuːɪti/ | 间断点 | jiàn duàn diǎn |
| removable discontinuity/rɪˈmuːvəbl dɪskɒntɪˈnjuːɪti/ | 可去间断点 | kě qù jiàn duàn diǎn |
| jump discontinuity/dʒʌmp dɪskɒntɪˈnjuːɪti/ | 跳跃间断点 | tiào yuè jiàn duàn diǎn |
| infinite discontinuity/ˈɪnfɪnət dɪskɒntɪˈnjuːɪti/ | 无穷间断点 | wú qióng jiàn duàn diǎn |
| vertical asymptote/ˈvɜːtɪkl ˈæsɪmptəʊt/ | 竖直渐近线 | shù zhí jiàn jìn xiàn |
Where a graph "breaks"
- A continuous curve can be drawn without lifting your pen. A discontinuity 间断点 is a break in that flow.
- Not all breaks are the same — there are three distinct types, each with its own graph signature.
- Naming the type tells you instantly whether the break can be repaired.
- Let's meet all three: removable, jump, and infinite.
图像在哪里"断裂"
- 一条连续曲线可以一笔画完、不抬笔。间断就是这种流畅中的断裂。
- 并非所有断裂都相同——有三种不同的类型,每种都有自己的图像特征。
- 说出类型,就立刻知道这个断裂能否修复。
- 我们来认识全部三种:可去、跳跃、无穷。
Removable: a single missing point
- A removable discontinuity 可去间断点 is just a hole in an otherwise smooth curve.
- The limit exists (both sides meet at one height), but $f(c)$ is missing or plotted elsewhere.
- $f(x)=\dfrac{x^2-4}{x-2}$ has a hole at $x=2$: the limit is $4$, yet $f(2)$ is undefined.
- Because the limit exists, you could "fill the hole" — hence removable (lesson 1.13).
可去:缺了一个点
- 可去间断只是一条本来平滑的曲线上的一个洞。
- 极限存在(两侧在同一高度相遇),但 $f(c)$ 缺失或被画在别处。
- $f(x)=\dfrac{x^2-4}{x-2}$ 在 $x=2$ 有一个洞:极限是 $4$,而 $f(2)$ 没有定义。
- 因为极限存在,你可以"把洞填上"——所以叫可去(第 1.13 课)。
At $x=3$, $f(x)=\dfrac{x^2-9}{x-3}$ has a limit of $6$ but $f(3)$ undefined. The discontinuity is... · 在 $x=3$ 处，$f(x)=\dfrac{x^2-9}{x-3}$ 的极限为 $6$ 但 $f(3)$ 未定义。该不连续性是...
A limit exists but the value is missing — a hole, i.e. removable. · 极限存在但函数值缺失 — 即空洞，属于可去不连续性。
Jump: the sides disagree
- A jump discontinuity 跳跃间断点 happens when the left and right limits are both finite but different.
- The graph leaps from one height to another — think of a step or a tax bracket.
- Here $\displaystyle\lim_{x\to c}f(x)$ does not exist, because the one-sided limits disagree.
- No single point value can bridge a gap of nonzero width, so a jump is not removable.
跳跃:两侧不一致
- 跳跃间断发生在左右极限都有限但不同时。
- 图像从一个高度跳到另一个高度——想想台阶或税率档次。
- 此时 $\displaystyle\lim_{x\to c}f(x)$ 不存在,因为两个单侧极限不一致。
- 没有哪个单点取值能跨过一个非零宽度的缺口,所以跳跃不可去。
A curve that blows up · 趋于无穷大的曲线
y = a / (x − b)
A reciprocal shoots to $\pm\infty$ near where its denominator hits zero — that is an infinite discontinuity and a vertical asymptote. · 倒数函数在其分母为零的点附近趋向于 $\pm\infty$ — 这是无穷间断点和垂直渐近线。
A jump discontinuity can be removed by redefining the function at the single point. · 跳跃间断点可以通过重新定义单点处的函数值来消除。
The one-sided limits differ, so no single value bridges the gap — jumps are not removable. · 单侧极限不同，因此没有单一值能填补间隙 — 跳跃不连续性不可消除。
A step function is $1$ for $x<0$ and $4$ for $x\ge0$. What is the size of the jump at $x=0$? · 阶跃函数是 $1$ ，对于 $x<0$ 和 $4$ 对于 $x\ge0$。在 $x=0$?
Right minus left: $4-1=3$. · 右减左：$4-1=3$。
Infinite: off to $\pm\infty$
- An infinite discontinuity 无穷间断点 is where the function grows without bound near $c$.
- The graph rockets up or down along a vertical asymptote — as in $f(x)=\dfrac1{x-2}$ at $x=2$.
- The limit is infinite (or DNE), so this break is also not removable.
- Signature: a denominator heading to zero while the numerator does not.
无穷:冲向 $\pm\infty$
- 无穷间断是函数在 $c$ 附近无界增长之处。
- 图像沿着一条竖直渐近线冲上或冲下——如 $f(x)=\dfrac1{x-2}$ 在 $x=2$。
- 极限是无穷(或不存在),所以这种断裂也不可去。
- 特征:分母奔向零而分子不为零。
An infinite discontinuity appears on a graph as a vertical ____. · 无穷间断点在图形上表现为垂直 ____。
The function grows without bound along the vertical asymptote. · 函数沿垂直渐近线无限增大。
Which discontinuity type is the only one where the two-sided limit exists? · 哪种不连续性类型是唯一一种双侧极限存在的情况？
Removable = a hole where both sides agree, so the limit exists. · 可去不连续性 = 两侧 agreeing 的空洞，因此极限存在。
Select all · 所有 functions with an infinite discontinuity at the marked point. · 选择所有在标记点处具有无穷不连续性的函数。
The third is removable (limit $=2$). The others blow up to $\pm\infty$. · 第三个是可去的（极限 $=2$）。其他的趋向于 $\pm\infty$。
Only the removable type has a genuine two-sided limit. For jump and infinite discontinuities the limit does not exist, so no clever redefinition at the single point can make the function continuous. Don't try to "fill" a jump — the gap has real width.
只有可去类型才有真正的双侧极限。对于跳跃和无穷间断,极限不存在,所以在那单点上再巧妙的重新定义也无法让函数连续。别想"填平"一个跳跃——那个缺口有真实的宽度。
Classify the discontinuity of each at the marked point:
- $\dfrac{x^2-9}{x-3}$ at $x=3$: limit $=6$, value undefined → removable (hole).
- $f(x)=\begin{cases}1,&x;<0\\2,&x;\ge0\end{cases}$ at $x=0$: left $\to1$, right $\to2$ → jump.
- $\dfrac{1}{x^2}$ at $x=0$: grows to $+\infty$ → infinite.
判断各函数在标记点处的间断类型:
- $\dfrac{x^2-9}{x-3}$ 在 $x=3$:极限 $=6$,取值无定义 → 可去(洞)。
- $f(x)=\begin{cases}1,&x;<0\\2,&x;\ge0\end{cases}$ 在 $x=0$:左 $\to1$,右 $\to2$ → 跳跃。
- $\dfrac{1}{x^2}$ 在 $x=0$:增长到 $+\infty$ → 无穷。
Three discontinuity types: removable (a hole — the limit exists but $f(c)$ doesn't match), jump (finite one-sided limits that disagree), and infinite (unbounded near $c$, a vertical asymptote). Only the removable type has a two-sided limit, so only it can be repaired.
三种间断类型:可去(一个洞——极限存在但 $f(c)$ 不匹配)、跳跃(有限的单侧极限但不一致)、无穷(在 $c$ 附近无界,一条竖直渐近线)。只有可去类型有双侧极限,所以只有它能被修复。