Intermediate Value Theorem
A continuous function maps the interval onto an interval: for every with there exists with .
Theorem 2.22 (Intermediate Value Theorem)
If is continuous, then is an interval. Hence, if , then there exists such that .
Proof
If is constant, or or , the claim is immediate. Otherwise assume WLOG and let . Then , so is non-empty, and is bounded above by , so exists in .
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If , set . By continuity of there is such that with gives , so and . Thus , contradicting the definition of as the least upper bound of .
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If , set . The same continuity argument gives , so in particular and hence , again contradicting the choice of .
Therefore .
Existence of N-th roots
For and , consider the continuous function
Since , the Intermediate Value Theorem gives with , that is, : a positive -th root of exists.
Related
Stated in
- Theorem 2.22 (Intermediate Value Theorem)§2.4 Intermediate Value Theorem