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Location of Roots Theorem

From AoPSWiki

The location of roots theorem is one of the most intutively obvious properties of continuous functions, as it states that if a continuous function attains positive and negative values, it must have a root (i.e. it must pass through 0).

Statement

Let f:[a,b]\rightarrow\mathbb{R} be a continuous function such that f(a)<0 and f(b)>0. Then there is some c\in (a,b) such that f(c)=0.

Proof

Let A=\{x|x\in [a,b],\; f(x)<0\}

As a\in A, A is non-empty. Also, as A\subset [a,b], A is bounded

Thus A has a least upper bound, \begin{align}\sup A& =u\in A.\end{align}

If f(u)<0:

As f is continuous at u, \exists\delta>0 such that x\in V_{\delta}(u)\implies f(x)<0, which contradicts (1).

Also if f(u)>0:

f is continuous imples \exists\delta>0 such that x\in V_{\delta}(u)\implies f(x)>0, which again contradicts (1) by the Gap lemma.

Hence, f(u)=0.

See Also

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.
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