riemann integrable

bigb · #1 $Old$ December 3rd, 2008, 03:58 PM

Suppose that f is continuous on [a,b], that f(x)≥0 for all x within [a,b] and that

. Prove that f(x)=0 for all x within [a,b]. Also, show that the continuity hypothesis cannot be dropped.

Plato · #2 $Old$ December 3rd, 2008, 04:24 PM

Here is a hint.
If $\left( {\exists c \in [a,b]} \right)\left[ {f(c) > 0} \right]$ then $\left( {\exists [d,e] \subseteq [a,b]} \right)\left( {\forall z \in [d,e]} \right)\left[ {f(z) > \frac{{f(c)}}{2}} \right]$ .
Ask yourself what is the minimum value of $\int_d^e {f(x)dx}$ .

Now drop continuity: $f(x) = \left\{ {\begin{array}{rl} {1/n} & {x = a + 1/n} \\ 0 & {else} \\ \end{array} } \right.$ .

bigb · #3 $Old$ December 4th, 2008, 09:29 AM

Quote:

Originally Posted by Plato $View Post$

Here is a hint.
If $\left( {\exists c \in [a,b]} \right)\left[ {f(c) > 0} \right]$ then $\left( {\exists [d,e] \subseteq [a,b]} \right)\left( {\forall z \in [d,e]} \right)\left[ {f(z) > \frac{{f(c)}}{2}} \right]$ .
Ask yourself what is the minimum value of $\int_d^e {f(x)dx}$ .

Now drop continuity: $f(x) = \left\{ {\begin{array}{rl} {1/n} & {x = a + 1/n} \\ 0 & {else} \\ \end{array} } \right.$ .

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