# question in sup of two functions...

• Apr 11th 2013, 06:23 AM
orir
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
• Apr 11th 2013, 08:25 AM
Plato
Re: question in sup of two functions...
Quote:

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.
• Apr 11th 2013, 08:50 AM
orir
Re: question in sup of two functions...
thanks..
but, what is High point theorem?
• Apr 11th 2013, 09:08 AM
Plato
Re: question in sup of two functions...
Quote:

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 $f(h)=\sup(f(a,b]))$.
• Apr 11th 2013, 09:12 AM
orir
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?