Darboux integral epsilon-delta proof in piecewise continuous function Using Darboux sums, if $f$ is a piecewise continuous function in $[a,b]$, then

It is integrable in $[a,b]$.

Given $\epsilon&gt;0$, there is $\delta&gt;0$ such that for every $P$ partition:
$$||P|| &lt; \delta\implies S\left(f,P\right) - I\left(f,P\right) &lt; \epsilon$$


I already know how to prove the first proposition, my question is for the second one. Please and thank you. This can be proven adding a point $c$ to $[a,b]$.
If $f$ is continuous then for any $\epsilon&gt; 0$ there exists a $\delta$ such that for all $(x,y)$ if $|x-y| &lt; \delta \implies |f(x) - f(y)|&lt; \epsilon.$

$\|P\|&lt; \delta \implies |x-y|&lt;\delta$

$|f(x) - f(y)|&lt; \epsilon$ for all $(x,y)$ in $P$ is an equivalent statement as $|\sup(f(x)) - \inf(f(x))|&lt; \epsilon$ for all $x$ in $P.$
If $f$ is continuous then for any $\epsilon&gt; 0$ there exists a $\delta$ such that for all $(x,y)$ if $|x-y| &lt; \delta \implies |f(x) - f(y)|&lt; \epsilon.$

$\|P\|&lt; \delta \implies |x-y|&lt;\delta$

$|f(x) - f(y)|&lt; \epsilon$ for all $(x,y)$ in $P$ is an equivalent statement as $|\sup(f(x)) - \inf(f(x))|&lt; \epsilon$ for all $x$ in $P.$