Suppose that f is continuous on [a,b], that f(x)≥0 for all x within [a,b] and that http://www.cramster.com/Answer-Board...8125008301.gif. Prove that f(x)=0 for all x within [a,b]. Also, show that the continuity hypothesis cannot be dropped.

Printable View

- Dec 3rd 2008, 01:58 PMbigbriemann integrable
Suppose that f is continuous on [a,b], that f(x)≥0 for all x within [a,b] and that http://www.cramster.com/Answer-Board...8125008301.gif. Prove that f(x)=0 for all x within [a,b]. Also, show that the continuity hypothesis cannot be dropped.

- Dec 3rd 2008, 02:24 PMPlato
Here is a hint.

If $\displaystyle \left( {\exists c \in [a,b]} \right)\left[ {f(c) > 0} \right]$ then $\displaystyle \left( {\exists [d,e] \subseteq [a,b]} \right)\left( {\forall z \in [d,e]} \right)\left[ {f(z) > \frac{{f(c)}}{2}} \right]$.

Ask yourself what is the minimum value of $\displaystyle \int_d^e {f(x)dx} $.

Now drop continuity: $\displaystyle f(x) = \left\{ {\begin{array}{rl}

{1/n} & {x = a + 1/n} \\

0 & {else} \\

\end{array} } \right. $. - Dec 4th 2008, 07:29 AMbigb