1. ## Continuous Functions

This question seems fairly logical if you think about it as the two functions must intersect. How would you go about proving it?

$\displaystyle Suppose\ that\ f\ and\ g\ are\ continuous$ $\displaystyle functions\ on\ the\ closed\ interval\ [a,b]$. $\displaystyle Show\ that\ there\ exists\ a\ point\ c \in [a,b]$ $\displaystyle such\ that\ f(c)=g(c)\ if\ f(a) \leq g(a)\ and\ f(b) \geq g(b)$

2. Originally Posted by acevipa
This question seems fairly logical if you think about it as the two functions must intersect. How would you go about proving it?

$\displaystyle Suppose\ that\ f\ and\ g\ are\ continuous$ $\displaystyle functions\ on\ the\ closed\ interval\ [a,b]$. $\displaystyle Show\ that\ there\ exists\ a\ point\ c \in [a,b]$ $\displaystyle such\ that\ f(c)=g(c)\ if\ f(a) \leq g(a)\ and\ f(b) \geq g(b)$
Look at h(x)= f(x)- g(x).

3. Originally Posted by HallsofIvy
Look at h(x)= f(x)- g(x).
Sorry, I still don't quite understand.

4. What is the sign of h(a)? What is the sign of h(b)?

5. Originally Posted by Defunkt
What is the sign of h(a)? What is the sign of h(b)?
Ok I see, so h(c)=0 as intercepts the x-axis. But how would you go about proving it still?

6. Would you say $\displaystyle h(a) \leq 0\ and h(b) \geq 0$

$\displaystyle Therefore,\ there\ exists\ a\ c \in (a,b)\ such\ that\ h(a)=0$

7. Yep, that is exactly the intermediate value theorem: Intermediate value theorem - Wikipedia, the free encyclopedia
Note, though, that since you have a weak inequality, it is possible that $\displaystyle c \in [a, b]$ ie. $\displaystyle c=a$ or $\displaystyle c=b$.