Is the Riemann integral of a strictly positive function positive? In the proof here a strictly positive function in $(0,\pi)$ is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a continuous function's graph lying entirely above the x-axis and the x-axis should not be zero. But how can I prove this formally?
If the function is positive over a closed interval apparently the result is not true (link goes to page 147 in Theories of Integration by Kurtz and Swarz). This has further confused me. Can someone please clarify my doubt.
Thanks

Here is a "direct" proof that does not require knowledge of the continuity properties of Riemann-integrable functions:


*

*Suppose by way of contradiction that $\int_0^1 f = 0$.

*It follows that the upper Darboux sums can be made arbitrarily small.  Choose some partition $\mathcal{P}_1$ such that $U(f; \mathcal{P}_1) < 1$.

*There must be some subinterval $I_1$ of $\mathcal{P}_1$ on which the supremum of $f$ is less than 1, as otherwise, summing over all subintervals contradicts the condition that $U(f; \mathcal{P}_1)< 1$.

*Choose some partition $\mathcal{P}_2$ for which $U(f; \mathcal{P}_2) < \frac{|I_1|}{2}$.  We may assume WLOG that $\mathcal{P}_2$ is a refinement of $\mathcal{P}_1$, as otherwise re replace $\mathcal{P}_2$ by their common refinement.

*Of the subintervals of $I_1$ which are in $\mathcal{P}_2$, their must be one, which we call $I_2$, on which the supremum of $f$ is less than $\frac{1}{2}$.  If not, the sum of $M_k \Delta x_k$ over the subintervals of $I_1$ would yield at least $\frac{|I_1|}{2}$, and $U(f; \mathcal{P}_2)$ would be at least this much.

*Similarly, choose $\mathcal{P}_3$ a refinement of $\mathcal{P}_2$ with the property $U(f; \mathcal{P}_3) < \frac{|I_2|}{3}$.

*Continuing in this manner, we obtain a nested sequence $I_1 \supseteq I_2 \supseteq \dots$ of closed bounded intervals with the property that $f(x) < \frac{1}{n}$ for all $x \in I_n$.

*By the Nested Interval Property, the intersection $\bigcap_{n=1}^\infty I_n$ is nonempty.  But any point $x$ in this intersection must have the property $\forall n \in \mathbb{N}: f(x) < \frac{1}{n}$, which implies $f(x) = 0$.

