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.
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. $.