http://www.ck12.org Chapter 14. Concepts of Calculus
Example A
Show that the converse of the Intermediate Value Theorem is false.
Solution:The converse of the Intermediate Value Theorem is:If there exists a value cā[a,b]such thatf(c) =ufor
everyubetweenf(a)andf(b)then the function is continuous.
In order to show the statement is false, all you need is one counterexample where every intermediate value is hit and
the function is discontinuous.
This function is discontinuous on the interval[ 0 , 10 ]but every intermediate value between the first height at( 0 , 0 )and
the height of the last point( 10 , 5 )is hit.