CALCULUS Understanding Its Concepts and Methods

Intermediate value theorem

Intermediate value
theoremLet
*f*
be continuous on the closed interval
[*a*,*b*].
If
*d*
is any number between
*f*(*a*)
and
*f*(*b*),
then
*d* = *f*(*c*)
for some number
*c*
between
*a*
and
*b*.

The intermediate value theorem says that there are no gaps in the set of values of a continuous function. Geometrically, the intermediate value theorem says that if the graph of a continuous function goes from one side of a horizontal line to the other, then it must cross the line somewhere.

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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.