# One Question In Real Analysis ..

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

"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