# integrbility proof

• Aug 28th 2011, 02:22 AM
transgalactic
integrbility proof
% Preview source code from paragraph 0 to 12
there is integrabile function f in {[}a,b{]} so f(a)<0.
f is continues from the right of a.
prove that there is $\displaystyle c\in(a,b)$ so $\displaystyle \intop_{a}^{c}f(x)dx<0$
prove:
there is $\displaystyle \epsilon=\frac{{|f(a)|}}{2}>0$ from the given continuety
in a there is a delta
$\displaystyle b-a>\delta>0$ so for every x in which $\displaystyle a\leq x<x+\delta$ $\displaystyle |f(x)-f(a)|<\varepsilon$.
so $\displaystyle f(x)-f(a)<-\frac{f(a)}{2}$ and the ({*}) $\displaystyle f(x)<f(a)/2<0$
f is integrabile in [a,b] and so in [a,a+$\displaystyle \delta/2$]
from ({*}) and from the monotonicity we conclude
$\displaystyle \intop_{a}^{a+\frac{\delta}{2}}f(t)dt\leq\intop_{a }^{a+\frac{\delta}{2}}\frac{f(a)}{2}dt=\frac{\delt a}{2}\frac{f(a)}{2}<0$
.
i got stuck on the first step:
why they chose such delta and x??
from the defintion it should be there is delta>0 for which in every
x a<x<a+$\displaystyle \delta$
• Aug 28th 2011, 02:25 AM
anonimnystefy
Re: integrbility proof
• Aug 28th 2011, 03:45 AM
transgalactic
Re: integrbility proof
it works now
• Aug 28th 2011, 03:55 AM
Plato
Re: integrbility proof
Quote:

Originally Posted by transgalactic
% Preview source code from paragraph 0 to 12
there is integrabile function f in {[}a,b{]} so f(a)<0.
f is continues from the right of a.
prove that there is $\displaystyle c\in(a,b)$ so $\displaystyle \intop_{a}^{c}f(x)dx<0$
prove:
there is $\displaystyle \epsilon=\frac{{|f(a)|}}{2}>0$ from the given continuety
in a there is a delta
$\displaystyle b-a>\delta>0$ so for every x in which $\displaystyle a\leq x<x+\delta$ $\displaystyle |f(x)-f(a)|<\varepsilon$.
so $\displaystyle f(x)-f(a)<-\frac{f(a)}{2}$ and the ({*}) $\displaystyle f(x)<f(a)/2<0$
f is integrabile in [a,b] and so in [a,a+$\displaystyle \delta/2$]
from ({*}) and from the monotonicity we conclude
$\displaystyle \intop_{a}^{a+\frac{\delta}{2}}f(t)dt\leq\intop_{a }^{a+\frac{\delta}{2}}\frac{f(a)}{2}dt=\frac{\delt a}{2}\frac{f(a)}{2}<0$
.
i got stuck on the first step:
why they chose such delta and x??
from the defintion it should be there is delta>0 for which in every
x a<x<a+$\displaystyle \delta$

"why they chose such delta and x??"
That is directly from the given: $\displaystyle f$ is continuous on the right at $\displaystyle a$.
We want an open interval on which $\displaystyle f$ is negative, so the integral will be negative.