# riemann integrable

• Dec 3rd 2008, 01:58 PM
bigb
riemann 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 PM
Plato
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 AM
bigb
Quote:

Originally Posted by Plato
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: $\displaystyle f(x) = \left\{ {\begin{array}{rl} {1/n} & {x = a + 1/n} \\ 0 & {else} \\ \end{array} } \right.$.

Tired messing around with your hint but could not get anywehre.