Xác nhận Tính liên tục trên Một Khoảng

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.

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: $\displaystyle\lim_{x\to b^-}f(x)=f(b)$.

Continuous on (a, b): no breaks inside and the one-sided limits match at the ends

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 $x>0$.

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.