Intermediate Value Theorem: Root Tests, Completeness, and Extensions

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Intermediate value theorem

If $f$ is continuous on $[a,b]$ and $u$ lies strictly between $f(a)$ and $f(b)$, then $f(c)=u$ for some $c\in(a,b)$.

Image of a continuous function on a closed interval

If $f$ is continuous on $[a,b]$, then $f([a,b])=[m,M]$, where $m$ and $M$ are its minimum and maximum there.

Is a sign change necessary for a continuous function to have a root?

No. For example, $f(x)=x^2$ has a root at $0$ without changing sign there.

Darboux function

A function that takes every value between its outputs at any two inputs in its domain.

What does completeness provide in the supremum proof of the IVT?

A least upper bound for the nonempty, bounded-above set of inputs whose shifted function values are nonpositive.

Quiz questions

1. Why does a continuous function on a closed bounded interval have image $[m,M]$, where $m$ and $M$ are its minimum and maximum?Answer: The extreme value theorem gives attained endpoints, and the IVT fills every value between them.
2. Suppose $f$ is continuous on $[a,b]$ and $f(a)$ and $f(b)$ have opposite signs. What follows?Answer: There is some $c\in(a,b)$ such that $f(c)=0$.
3. Which property does Darboux's theorem guarantee for every derivative?Answer: It takes every intermediate value between any two values it attains.
4. Let $g(x)=f(x)-u$ be continuous on $[a,b]$, with $g(a)<0<g(b)$, and let $S=\{x\in[a,b]:g(x)\leq 0\}$. What does completeness ensure?Answer: The nonempty set $S$, bounded above, has a least upper bound in the real numbers.
5. If $X$ is connected and $f:X\to Y$ is continuous, what topological property must $f(X)$ have?Answer: The image $f(X)$ is connected.