Intermediate Value Theorem

romano · Riemann Hypothesis Offline Joined: 10 Jun 2004 Posts: 304

What is the Intermediate Value Theorem ?

**Posted:** Tue Jan 11, 2005 4:33 am

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Peter Scholze · Yang-Mills Theory Offline Joined: 03 Jun 2004 Posts: 674

if $f: I\to\mathbb{R}$ is a continuous function on some interval $I$ and if $a,b\in I$ and $f(a)\leq x\leq f(b)$ or $f(a)\geq x\geq f(b)$ , then there is a $a\leq \xi\leq b$ s.t. $f(\xi)=x$ .

**Posted:** Tue Jan 11, 2005 7:46 am

Valentin Vornicu

I have 2 observations for what Peter said:

1) The IVT starts from the following Property that is the Intermediate Value Property (aka IVP):
A function $f: I \to\mathbb{R}$ has the I.V.P. if and only if for all $a<b$ , $a,b \in I$ , and for all $\gamma \in (f(a),f(b))$ (if $f(a)>f(b)$ then the interval is $(f(b),f(a))$ ) there exists an $c \in (a,b)$ such that $f(c) = \gamma$ . This is also called (in Romania) the Darboux Property. I'm not sure if this is also an international notation.

This definition obviously implies that all continous functions have the IVP, however there are some (rather complicated) examples of non-continous, but IVP functions.

2) The most times this is used (in olympiad problems in general) is the following: if you have an polynomial $P(X) \in \mathbb{R}[X]$ , and for some two reals $a,b$ you find that $P(a)\cdot P(b)<0$ then the polynomial has a root in the interval $(a,b)$ .

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jhaussmann5 · Riemann Hypothesis Offline Joined: 29 Dec 2003 Posts: 343

Valentin,
I believe Peter's statement is correct: he is stating the Intermediate Value Theorem, you are stating the Intermediate Value Property.

Thus, the theorem is that all continuous functions satisfy the IVP. See, e.g. www.math.uncc.edu/~hbreiter/m1120/lectures/lect16.htm

**Posted:** Tue Jan 11, 2005 4:34 pm

Valentin Vornicu

**Posted:** Tue Jan 11, 2005 4:53 pm

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We all use math everyday: to forecast weather, to tell time, to handle money; we also use math to analyze crime, reveal patterns, predict behavior. Using numbers we can solve the biggest mysteries we know.