# One Question In Real Analysis ..

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• March 24th 2010, 12:33 PM
Miral
One Question In Real Analysis ..
Hi all..

i want to ask you about this question in real analysis..

"suppose that f continuous on [a,b] , that f(x) >= 0 (more or equal 0 ) for all x in [a,b] and that http://up.v1va.com/files/xfhdxtawmu5q9lg5y8mf.jpg=0

prove that f(x)=0 for all x in [a,b] "
• March 24th 2010, 12:45 PM
Bruno J.
Suppose $\exists c \in [a,b]$ such that $f(c) > 0$. Then there exists a neighborhood $N$ of $c$ such that $f(x)>f(c)/2$ for all $x \in N \cap [a,b]$. (Prove this using the continuity of $f$ at $c$.) Then $0 = \int_a^b f \geq \int_{N \cap [a,b]} f(c)/2 > 0$ which is false.
• March 24th 2010, 01:41 PM
Miral
thanx for your answer..
but i want the solution in another way..
can i solve it by cauchy criteria and tagged partition or squeezing theorem..
im Really Confused and i cant join the information together..
• March 24th 2010, 01:51 PM
Bruno J.
I don't see how anything that you mention really has much to do with the problem. The solution I gave is quite standard : suppose the function is not identically zero, then it's bounded away from zero on a small subinterval, and hence its integral on this small subinterval is positive, and the integral on the whole interval is at least the integral on this small subinterval.