Intermediate Value Theorem

Core Sentence / 核心句

A continuous function on a closed interval must attain every output value between its endpoint outputs.

中文理解：连续函数不能从一个端点高度跳到另一个端点高度；中间的每个函数值都必须经过。

Hypothesis / 前提

$$f\text{ is continuous on }[a,b].$$

Conclusion / 结论

If $L$ lies between $f(a)$ and $f(b)$, then there exists $c\in [a,b]$ such that

$$f(c)=L.$$

Watch Out / 易错点

• IVT guarantees existence, not uniqueness.
• It does not locate $c$.
• It does not require differentiability.
• It concerns function values, not derivative values.

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第5次正课 · MVT Versus IVT
  • IVT guarantees an intermediate output under continuity, while MVT guarantees a derivative value under stronger hypotheses

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Extreme Value Theorem
• 微积分BC第5次正课-相关变化率线性近似与中值定理
• Mean Value Theorem
• Limits
• Derivatives
• 微积分BC第1次正课-极限与连续