Continuous Functions

• Apr 28th 2010, 02:08 AM
acevipa
Continuous Functions
How would I go about doing this question:

$Suppose\ that\ f\ and\ g$ $are\ continuous\ functions\ on\ the\ closed\ interval$ $[0,1]\ and\ that\ 0 \leq f(x) \leq 1\ for\ every\ x\ in\ [0,1].$ $Show\ that\ there\ exists\ a\ real\ number$ $c \in [0,1]\ such\ that\ f(c)=c.$

Would you do this question using the intermediate value theorem to $g(x)=f(x)-x$
• Apr 28th 2010, 02:19 AM
HallsofIvy
Quote:

Originally Posted by acevipa
How would I go about doing this question:

$Suppose\ that\ f\ and\ g$ $are\ continuous\ functions\ on\ the\ closed\ interval$ $[0,1]\ and\ that\ 0 \leq f(x) \leq 1\ for\ every\ x\ in\ [0,1].$ $Show\ that\ there\ exists\ a\ real\ number$ $c \in [0,1]\ such\ that\ f(c)=c.$

Would you do this question using the intermediate value theorem to $g(x)=f(x)-x$

Yes, that's a good idea. I suggest you look at g(0) and g(1).
• Apr 28th 2010, 02:29 AM
acevipa
Quote:

Originally Posted by HallsofIvy
Yes, that's a good idea. I suggest you look at g(0) and g(1).

Would you do it like this

$\frac{g(1)-g(0)}{1-0} = g'(c)$

$\frac{f(1)-1-f(0)+0}{1-0}=f'(c)-1$

$Given\ that\ f(1)=1\ and\ f(0)=0$

$\frac{1-1-0}{1}=f'(c)-1$

$\frac{0}{1}=f'(c)-1$

$f'(c)-1=0$

$f'(c)=1\ \forall c \in (0,1)$

What do I do from here?