Math 2.0

Real functions Theorem

Intermediate value theorem

Builds on Continuous function

Statement

Let a<ba < b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \mathbb{R} be continuous. Let yy be a real number between f(a)f(a) and f(b)f(b). Then there is c∈[a,b]c \in [a, b] with f(c)=yf(c) = y.

Remarks

Assume f(a)<y<f(b)f(a) < y < f(b). Let cc be the supremum of {x∈[a,b]:f(x)<y}\{ x \in [a, b] : f(x) < y \}. Continuity gives f(c)≤yf(c) \le y and f(c)≥yf(c) \ge y. The proof needs the completeness of ℝ\mathbb{R}. Over ℚ\mathbb{Q}, the function x2−2x^2 - 2 changes sign on [1,2][1, 2] but has no rational zero. It gives the existence of nn-th roots, and it shows that each odd degree real polynomial has a real root. Mathlib states it as intermediate_value_Icc.

History

Bernard Bolzano first proved the theorem in 1817.

Reading path

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionIntermediate value theorem, statement, no evidence levelIntermediate value th…statement
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  1. Limit point of a set definition
  2. Limit of a function definition
  3. Continuity at a point definition
  4. Continuous function definition
  5. Intermediate value theorem statement

Where this leads

86 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Intermediate_value_theorem, Theorem
About this page
Tag00KAFBR32E
URIm2://00KAFBR32E
Aliasesbolzano-theorem
Kindstatement, theorem
WritingW1 stated
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 00KAFBR32E.
UsesContinuous function
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