Converse Is False
A "Darboux function" is a real-valued function f that has the "intermediate value property", i.e., that satisfies the conclusion of the intermediate value theorem: for any two values a and b in the domain of f, and any y between f(a) and f(b), there is some c between a and b with 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 f : defined by f(x) = sin(1/x) for x > 0 and f(0) = 0. This function is not continuous at x = 0 because the limit of f(x) as 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.
Historically, this intermediate value property has been suggested as a definition for continuity of real-valued functions; this definition was not adopted.
Darboux's theorem states that all functions that result from the differentiation of some other function on some interval have the intermediate value property (even though they need not be continuous).
Read more about this topic: Intermediate Value Theorem
Famous quotes containing the words converse is, converse and/or false:
“It is said that desire is a product of the will, but the converse is in fact true: will is a product of desire.”
—Denis Diderot (17131784)
“Lately in converse with a New York alec
About the new school of the pseudo-phallic ...”
—Robert Frost (18741963)
“A good face they say, is a letter of recommendation. O Nature, Nature, why art thou so dishonest, as ever to send men with these false recommendations into the World!”
—Henry Fielding (17071754)