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The equivalence between this formulation and the modern one can be shown by setting <math>\phi</math> to the appropriate constant function. [[Augustin-Louis Cauchy]] provided the modern formulation and a proof in 1821.<ref name="grabiner">{{Cite journal| title=Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus| first=Judith V.| last=Grabiner| journal=The American Mathematical Monthly| date=March 1983| volume=90| pages=185–194| url=http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Grabiner185-194.pdf| doi=10.2307/2975545| issue=3| jstor=2975545}}</ref> Both were inspired by the goal of formalizing the analysis of functions and the work of [[Joseph-Louis Lagrange]]. The idea that continuous functions possess the intermediate value property has an earlier origin. [[Simon Stevin]] proved the intermediate value theorem for [[polynomial]]s (using a [[Cubic function|cubic]] as an example) by providing an algorithm for constructing the decimal expansion of the solution. The algorithm iteratively subdivides the interval into 10 parts, producing an additional decimal digit at each step of the iteration.<ref>Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. [[Foundations of Science]]. {{doi|10.1007/s10699-011-9223-1}} See [https://doi.org/10.1007%2Fs10699-011-9223-1 link]</ref> Before the formal definition of continuity was given, the intermediate value property was given as part of the definition of a continuous function. Proponents include [[Louis Arbogast]], who assumed the functions to have no jumps, satisfy the intermediate value property and have increments whose sizes corresponded to the sizes of the increments of the variable.<ref>{{MacTutor Biography|id=Arbogast}}</ref>
Earlier authors held the result to be intuitively obvious and requiring no proof. The insight of Bolzano and Cauchy was to define a general notion of continuity (in terms of [[infinitesimal]]s in Cauchy's case and using real inequalities in Bolzano's case), and to provide a proof based on such definitions.
==Converse is false==▼
A [[Darboux function]] is a real-valued function {{mvar|f}} that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values {{mvar|a}} and {{mvar|b}} in the ___domain of {{mvar|f}}, and any {{mvar|y}} between {{math|''f''(''a'')}} and {{math|''f''(''b'')}}, there is some {{mvar|c}} between {{mvar|a}} and {{mvar|b}} with {{math|1=''f''(''c'') = ''y''}}. The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.▼
As an example, take the function {{math|''f'' : [0, ∞) → [−1, 1]}} defined by {{math|1=''f''(''x'') = sin(1/''x'')}} for {{math|''x'' > 0}} and {{math|1=''f''(0) = 0}}. This function is not continuous at {{math|1=''x'' = 0}} because the [[limit of a function|limit]] of {{math|1=''f''(''x'')}} as {{mvar|x}} tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the [[Conway base 13 function]].▼
In fact, [[Darboux's theorem (analysis)|Darboux's theorem]] states that all functions that result from the [[derivative|differentiation]] of some other function on some interval have the [[intermediate value property]] (even though they need not be continuous).▼
Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions;<ref>{{Cite book |last=Smorynski |first=Craig |url=https://books.google.com/books?id=lnuhDgAAQBAJ&pg=PA51&q=Historically%2C+this+intermediate+value+property+has+been+suggested+as+a+definition+for+continuity+of+real-valued+functions |
==Generalizations==
=== Multi-dimensional spaces ===
The [[Poincaré-Miranda theorem]] is a generalization of the Intermediate value theorem from a (one-dimensional) interval to a (two-dimensional) rectangle, or more generally, to an n-dimensional [[N-cube|cube]].
=== General metric and topological spaces ===
The intermediate value theorem is closely linked to the [[topology|topological]] notion of [[Connectedness (topology)|connectedness]] and follows from the basic properties of connected sets in metric spaces and connected subsets of '''R''' in particular:
* If <math>X</math> and <math>Y</math> are [[metric space]]s, <math>f \colon X \to Y</math> is a continuous map, and <math>E \subset X</math> is a [[Connected space|connected]] subset, then <math>f(E)</math> is connected. <math>(*)</math>
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The [[Brouwer fixed-point theorem]] is a related theorem that, in one dimension, gives a special case of the intermediate value theorem.
▲==Converse is false==
▲A [[Darboux function]] is a real-valued function {{mvar|f}} that has the "intermediate value property," i.e., that satisfies the conclusion of the intermediate value theorem: for any two values {{mvar|a}} and {{mvar|b}} in the ___domain of {{mvar|f}}, and any {{mvar|y}} between {{math|''f''(''a'')}} and {{math|''f''(''b'')}}, there is some {{mvar|c}} between {{mvar|a}} and {{mvar|b}} with {{math|1=''f''(''c'') = ''y''}}. The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false.
▲As an example, take the function {{math|''f'' : [0, ∞) → [−1, 1]}} defined by {{math|1=''f''(''x'') = sin(1/''x'')}} for {{math|''x'' > 0}} and {{math|1=''f''(0) = 0}}. This function is not continuous at {{math|1=''x'' = 0}} because the [[limit of a function|limit]] of {{math|1=''f''(''x'')}} as {{mvar|x}} tends to 0 does not exist; yet the function has the intermediate value property. Another, more complicated example is given by the [[Conway base 13 function]].
▲In fact, [[Darboux's theorem (analysis)|Darboux's theorem]] states that all functions that result from the [[derivative|differentiation]] of some other function on some interval have the [[intermediate value property]] (even though they need not be continuous).
▲Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions;<ref>{{Cite book|url=https://books.google.com/books?id=lnuhDgAAQBAJ&pg=PA51&q=Historically%2C+this+intermediate+value+property+has+been+suggested+as+a+definition+for+continuity+of+real-valued+functions | title=MVT: A Most Valuable Theorem|last=Smorynski|first=Craig|date=2017-04-07|publisher=Springer| isbn=9783319529561| language=en}}</ref> this definition was not adopted.
==In constructive mathematics==
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==See also==
* {{annotated link|Mean value theorem}}
* {{annotated link|Non-atomic measure}}
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