Math Help - question in sup of two functions...

1. question in sup of two functions...

f,g: [a,b]---> R
f(x)>g(x) for every x at [a,b]

which of these statements are true? and why?
1. if f and g are continous at (a,b), and f is bounded at [a,b] - so sup f((a,b))>sup g((a,b)).
2. if f and g are continous at [a,b], so sup f([a,b])>sup g([a,b]).

thanks

2. Re: question in sup of two functions...

Originally Posted by orir
f,g: [a,b]---> R
f(x)>g(x) for every x at [a,b]
which of these statements are true? and why?
1. if f and g are continous at (a,b), and f is bounded at [a,b] - so sup f((a,b))>sup g((a,b)).
2. if f and g are continous at [a,b], so sup f([a,b])>sup g([a,b]).

For #1, let $[a,b]=[0,1]$ and consider $f(x)=x^3~\&~g(x) = \left\{ {\begin{array}{*{20}{rl}} {{x^2},}&{0 < x < 1} \\ { - 1,}&{x \in \left\{ {0,1} \right\}} \end{array}} \right.$
Can you show that $\sup(f([0,1])=\sup(g([0,1])~?$

For #2, think High point theorem.

3. Re: question in sup of two functions...

thanks..
but, what is High point theorem?

4. Re: question in sup of two functions...

Originally Posted by orir
what is High point theorem?

If $f$ is a continuous function on $[a,b]$, then $\exists h\in [a,b]$ such that $f(h)=\sup(f(a,b]))$.

5. Re: question in sup of two functions...

ok! so using this theorem 2# is easy.. thank you!
how can i find more details on this theorem? i couldn't find it by just searching "Hight point theorem" up... are there anymore usefull sentences like this one?