Given $f$ is Rimann integrable on $[a,b]$, and $f(x)$ is positive at $\forall x \in [a,b]$. Using Heine-Borel's finite covering theorem to prove $\int_a^b f(x) \,\mathrm{d} x>0$.
Prove $\int_a^b f(x)\,\mathrm dx>0$ if $f(x)>0$ at $\forall x\in [a,b]$ and Riemann integrable (via finite covering theorem).
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calculus
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1Consider $\inf _ {x \in [a, b]} f (x)$. Also, put some effort into your questions. – 2012-11-04
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0Consider the Riemann function $f(x)=1/q$, when $x=p/q$; and $f(x)=0$ when $x$ isn't a quotient number. We could find that $g(x)=f(x)+\epsilon$ is Riemann integrable on $[0,1]$ and $\inf_{x\in[a,b]} f(x)=0$, while $f(x)>0$ on $[0,1]$. So, I just don't know what @Karolis Juodelė want to say! – 2012-11-05