Editing Intermediate value theorem (section)

==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,&thinsp;∞) → [−1,&thinsp;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&q=Historically%2C+this+intermediate+value+property+has+been+suggested+as+a+definition+for+continuity+of+real-valued+functions&pg=PA51 |title=MVT: A Most Valuable Theorem |date=2017-04-07 |publisher=Springer |isbn=9783319529561 |language=en}}</ref> this definition was not adopted.