Prove that if $f$ is defined and bounded in $[a,b]$ and integrable in $[c,b]$ for all $c\in(a,b)$ then $f$ is integrable in $[a,b]$ Prove that if $f$ is defined and bounded in $[a,b]$ and integrable in $[c,b]$ for all $c\in(a,b)$ then $f$ is integrable in $[a,b]$. 
I don't even know where to begin - I've tried to show that for all $\varepsilon>0$ there exists a partition $P$ such that $S(P)-s(P) < \varepsilon$ but I haven't gotten very far. Any help would be appreciated.
 A: Since $f$ is bounded on $[a,b]$, for any $\epsilon>0$, there is a $c\in(a,b)$ so that $$ \bigl(\sup_{x\in[a,c]} f(x)- \inf_{x\in[a,c]}f(x)\bigr)\cdot (c-a)<\epsilon/2 .$$
Given $\epsilon>0$, first choose $c\in(a,b)$ small enough so that the above inequality holds. 
Then, since $f$ is integrable on $[c,b]$,  you may  find a partition, $P$, of $[c,b]$ such that $S(P)−s(P)<\epsilon/2$,  where $S(P)$ is the upper sum of $f$ corresponding to $P$ and $s(P)$ is the lower sum of $f$ corresponding to $P$. 
Now consider the partition $P_0= \{\,a\,\}\cup P$ of $[a,b]$. Note that 
$$ 
S(P_0)-s(P_0) = 
\bigl(\sup_{x\in[a,c]} f(x)- \inf_{x\in[a,c]}f(x)\bigr)\cdot (c-a)
+S(P)-s(P).
$$
I'll leave the rest for you ...
A: A first step in solving this is what you have already done.  Assume you are given $\varepsilon>0$. You want to find a $\delta>0$ such that your upper sums and lower sums differ by less than $\varepsilon$. 
Now translate the information you are given into a language of relevant symbols you will have to talk about. Ask questions about the meaning of the info given drawing pictures to help illustrate to you what these symbols mean.  For example:
What does it mean to say that $f$ is bounded on $[a,b]$? Give a name such as $B$ to some bound (it is neater but not necessary to bound the absolute value of $f$; you can use different upper and lower bounds). Draw a picture showing these are bounds.
What is a partition? Draw a picture to label everything related to the partition $P$ and say what its mesh is in terms of the labels. 
What does it mean to say for all $c\in (a,b)$, $f$ is integrable on $[c,b]$? Write down the $\varepsilon,\delta$ definition. 
In terms of the pictures you have drawn and the labels assigned to the partition, what is some $c\in(a,b)$ you might use to divide the interval $[a,b]$ into 2 subintervals on each of which you can bound the difference $S(P)-s(P)$ in terms of the symbols you have introduced. Remember that dividing by a nonzero number large in absolute value gives a number very small in absolute value.
This is the way almost all the proofs you see involving finding $\delta$ to make the absolute value of something  less that a given positive $\varepsilon$ begin. Don't worry if you get a $\delta$ giving a bound something like $m\epsilon$ for some positive $m$; you can always say you can do this starting with $\varepsilon/m$ so the result is proved. 
I hope this enables you to complete the proof.    
