# [SOLVED] integrability of f and f^2

• October 11th 2008, 12:33 AM
akolman
[SOLVED] integrability of f and f^2
I need help with this problem.

Let[tex] f(x)[\math] be a bounded function on $[a,b]$. Let $B>0$ and $|f(x)| \leq B$ for $\forall x \in [a,b]$

Show that $U(f^2,P)-L(f^2,P) \leq 2B[U(f,P)-L(f,P)]$ for all partitions P of [a,b]
• October 11th 2008, 03:28 AM
Opalg
Quote:

Originally Posted by akolman
Let $f(x)$ be a bounded function on $[a,b]$. Let $B>0$ and $|f(x)| \leq B$ for $\forall x \in [a,b]$

Show that $U(f^2,P)-L(f^2,P) \leq 2B[U(f,P)-L(f,P)]$ for all partitions P of [a,b]

Difference of two squares: $\left|f^2(x_1) - f^2(x_2)\right| = |f(x_1) + f(x_2)|\,|f(x_1) - f(x_2)|\leqslant2B|f(x_1) - f(x_2)|$.
• October 11th 2008, 06:59 AM
akolman
Ohh man! I can believe it! tank you very much!!(Clapping)