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

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 $\displaystyle [a,b]=[0,1]$ and consider $\displaystyle f(x)=x^3~\&~g(x) = \left\{ {\begin{array}{*{20}{rl}} {{x^2},}&{0 < x < 1} \\ { - 1,}&{x \in \left\{ {0,1} \right\}} \end{array}} \right.$

Can you show that $\displaystyle \sup(f([0,1])=\sup(g([0,1])~?$

For #2, think *High point theorem.*

Re: question in sup of two functions...

thanks..

but, what is High point theorem?

Re: question in sup of two functions...

Quote:

Originally Posted by

**orir** what is High point theorem?

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

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?