# Darboux integral epsilon-delta proof in piecewise continuous function

Using Darboux sums, if $$f$$ is a piecewise continuous function in $$[a,b]$$, then

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

2. Given $$\epsilon>0$$, there is $$\delta>0$$ such that for every $$P$$ partition:

$$||P|| < \delta\implies S\left(f,P\right) - I\left(f,P\right) < \epsilon$$

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 $$[a,b]$$.

• What is you definition of integrability? Do you use Darboux sum or Riemann sum to define integral? Moreover if you use Riemann sum then is the limit of Riemann sum based on letting norm $||P||$ of partition $P$ tending to $0$ or is it based on making partition $P$ finer and finer by adding more points of subdivision? Further what is $I(f, P)$? If you add these details then it is possible to give a precise answer. Jun 28 '16 at 14:26

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