Confirming Continuity over an Interval · 确认区间上的连续性
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| interval/ˈɪntəvl/ | 区间 | qū jiān |
| domains/dəˈmeɪnz/ | 定义域 | dìng yì yù |
From one point to a whole stretch
- Continuity at a single point is nice; usually we want it across an interval 区间.
- A function is continuous on an interval if it is continuous at every point inside it.
- That's what lets you draw the whole stretch without lifting your pen.
- Two subtleties: how to treat the endpoints, and which familiar functions are automatically continuous.
从一个点到一整段
- 在单个点连续很好;但我们通常想要在一整个区间上连续。
- 若函数在区间内的每一个点都连续,它就在这个区间上连续。
- 这才让你能把整段一笔画完、不抬笔。
- 有两个微妙之处:如何处理端点,以及哪些熟悉的函数自动连续。
Continuous across the whole view · 在整个视图范围内连续
y = ax³ + bx
A polynomial is continuous on every interval — no holes, jumps, or blow-ups anywhere you look. · 多项式在任何区间上都连续 — 无论你在哪里看都没有空洞、跳跃或爆炸。
Open vs. closed intervals
- On an open interval $(a,b)$: check the three-condition continuity at every interior point.
- On a closed interval $[a,b]$: also handle the endpoints, where you can only approach from one side.
- At the left end $a$, require right-continuity: $\displaystyle\lim_{x\to a^+}f(x)=f(a)$.
- At the right end $b$, require left-continuity: $\displaystyle\lim_{x\to b^-}f(x)=f(b)$.
开区间与闭区间
- 在开区间 $(a,b)$ 上:对每个内部点检查三条件连续性。
- 在闭区间 $[a,b]$ 上:还要处理端点,那里你只能从一侧靠近。
- 在左端 $a$,要求右连续:$\displaystyle\lim_{x\to a^+}f(x)=f(a)$。
- 在右端 $b$,要求左连续:$\displaystyle\lim_{x\to b^-}f(x)=f(b)$。
At the left · 左侧 endpoint $a$ of a closed interval $[a,b]$, continuity requires... · 在闭区间的 左 端点 $a$ 处，连续性要求... $[a,b]$，连续性要求……
You can only approach the left endpoint from the right, so right-continuity is required. · 你只能从右侧接近左端点，因此需要右连续。
For · 支持 $f(x)=\sqrt x$ on $[0,9]$, the left-continuity check at the right endpoint needs $\lim_{x\to9^-}\sqrt x$. What is that value? · 对于 $f(x)=\sqrt x$ 在 $[0,9]$ 上，右端点处的左连续性检查需要 $\lim_{x\to9^-}\sqrt x$。该值是什么？
$\sqrt 9 = 3$, which equals $f(9)$, so $f$ is left-continuous there. · $\sqrt 9 = 3$，它等于 $f(9)$，因此 $f$ 在该点左连续。
The functions that come "pre-continuous"
- On their natural domains 定义域, these families are continuous everywhere:
- Polynomials — continuous for all real $x$.
- Rational, exponential, logarithmic, and trigonometric functions — continuous wherever they are defined.
- So a rational function is continuous except where its denominator is zero; $\ln x$ is continuous for $x>0$.
"天生连续"的那些函数
- 在各自的自然定义域上,这些族处处连续:
- 多项式——对所有实数 $x$ 连续。
- 有理、指数、对数、三角函数——只要有定义就连续。
- 所以有理函数除了分母为零之处都连续;$\ln x$ 对 $x>0$ 连续。
Which function is continuous for all · 所有 real · 实数 $x$? · 哪个函数对所有实数 $x$ 都是连续的？
Polynomials are continuous everywhere; the others have domain gaps. · 多项式在全域连续；其他函数都有定义域缺口。
Select all · 所有 function families that are continuous on their entire natural domain. · 选择所有在其整个自然定义域上连续的所有函数族。
All four are continuous throughout their domains — the domain just excludes trouble points (e.g. rational at zeros of the denominator). · 这四种函数在其定义域内全部连续 — 定义域只是排除了麻烦点（例如分母为零处的有理函数）。
Why we care: it unlocks theorems
- Continuity on a closed interval is the entry ticket to the big theorems ahead.
- The Intermediate Value Theorem (1.16) and the Extreme Value Theorem (Unit 5) both require continuity on $[a,b]$.
- Skip the continuity check and those theorems simply do not apply — a common exam trap.
- So confirming continuity over an interval is not busywork; it is the hypothesis everything else stands on.
为何在意:它解锁定理
- 闭区间上的连续性是通往后面几大定理的入场券。
- 介值定理(1.16)和极值定理(第 5 单元)都要求在 $[a,b]$ 上连续。
- 跳过连续性检查,这些定理就根本不适用——一个常见的考试陷阱。
- 所以确认区间上的连续性不是无用功;它是其他一切赖以成立的前提。
$f(x)=\dfrac1x$ is continuous on the interval $[-1,1]$. · $f(x)=\dfrac1x$ 在区间 $[-1,1]$ 上连续。
$0\in[-1,1]$ and $f$ blows up there, so it is not continuous on that interval. · $0\in[-1,1]$ 和 $f$ 在那里爆炸，所以它在该区间上不连续。
Continuity on a closed interval is a required hypothesis of the ____ Value Theorem. · 闭区间上的连续性是 ____ 值定理的必要假设。
The IVT (and the Extreme Value Theorem) require continuity on $[a,b]$. · 介值定理（和极值定理）要求在 $[a,b]$ 上连续。
"Continuous on its domain" is not the same as "continuous everywhere." $f(x)=\dfrac1x$ is continuous on its domain (all $x\neq0$), yet it is not continuous on $[-1,1]$ because $0$ is in that interval and $f$ blows up there. Always check that the interval avoids the trouble spots.
"在定义域上连续"不等于"处处连续"。$f(x)=\dfrac1x$ 在其定义域(所有 $x\neq0$)上连续,却不在 $[-1,1]$ 上连续,因为 $0$ 在那个区间里,$f$ 在此爆发。永远检查区间是否避开了麻烦点。
Is $f(x)=\sqrt{x}$ continuous on $[0,4]$?
- Interior $(0,4)$: $\sqrt x$ is continuous (a standard function on its domain). ✓
- Left endpoint $0$: $\displaystyle\lim_{x\to0^+}\sqrt x=0=f(0)$ — right-continuous. ✓
- Right endpoint $4$: $\displaystyle\lim_{x\to4^-}\sqrt x=2=f(4)$ — left-continuous. ✓
- So $f$ is continuous on the closed interval $[0,4]$.
$f(x)=\sqrt{x}$ 在 $[0,4]$ 上连续吗?
- 内部 $(0,4)$:$\sqrt x$ 连续(在定义域上的标准函数)。✓
- 左端点 $0$:$\displaystyle\lim_{x\to0^+}\sqrt x=0=f(0)$——右连续。✓
- 右端点 $4$:$\displaystyle\lim_{x\to4^-}\sqrt x=2=f(4)$——左连续。✓
- 所以 $f$ 在闭区间 $[0,4]$ 上连续。
$f$ is continuous on an interval when it is continuous at every point inside, using one-sided continuity at the endpoints of a closed interval. Polynomials, rational, exponential, logarithmic, and trigonometric functions are continuous on their domains. This interval-continuity is the hypothesis the IVT and EVT depend on.
$f$ 在区间上连续,当它在内部每一点都连续,并在闭区间端点用单侧连续性。多项式、有理、指数、对数、三角函数在各自的定义域上连续。这种区间连续性正是介值定理与极值定理所依赖的前提。