Bounded and discontinuous proof Problem: Suppose that $f : [a, b] \to \mathbb{R}$ is bounded and discontinuous at exactly
one point $c$ between $a$ and $b$. Prove that $f$ is $\mathbb{R}$-integrable.
I know that $f$ is bounded and continuous on $(a,c)$ which implies that $f$ is integrable on $(a,c)$.
I also know that $f$ is bounded and continuous on $(b,c)$ which implies $f$ is integrable on $(b,c)$.
Therefore we know that $f$ is integrable on $(a,b)$ and $\int_a^b f(x)\:\mathrm{d}x = \int_a^c f(x)\:\mathrm{d}x+\int_c^b f(x)\:\mathrm{d}x$.
I don't feel like this really proves the problem for me. So what is missing or needs to be expanded on?
 A: I am guessing at what "previous" theorems you have to work with for this problem, but I suspect that you have something like
If $f$ is integrable on $(a,b)$ and $a < c < b$ then $f$ is integrable on $(a,c)$ and $(c,b)$ and in fact
$$\int_a^b f \, dx = \int_a^c f \, dx + \int_c^b f \, dx$$
But you may not use this identity to conclude that if the right hand side exists then so does the left.  In fact this is nearly the problem you are being asked to solve!
A: We can show that $f$ satisfies the Riemann criterion for integrability by enclosing the point of discontinuity -- which may not be  a removable or simple jump discontinuity -- in a sufficiently small interval.
Choose some $\delta > 0$ such that $(c-\delta, c+\delta) \subset [a,b]$. Then $f$ is integrable over $[a,c-\delta]$ and $[c+\delta,b].$
For any $\epsilon > 0$,  there are partitions $P'$ of $[a,c-\delta]$ and $P''$ of  $[c+\delta,b]$ such that $U(P',f) - L(P',f) < \epsilon/3$ and  $U(P'',f) - L(P'',f) < \epsilon/3$.
Form the partition $P = P' \cup \{c-\delta,c+\delta\} \cup P''$ of $[a,b].$
Then
$$U(P,f) - L(P,f) \\= [U(P',f) - L(P',f)] + [U(P'',f) - L(P'',f)] + 2\delta\sup_{x,y \in[c-\delta,c + \delta]}|f(x)-f(y)|\\\leqslant\frac{2\epsilon}{3}+(M-m)2\delta.$$
where 
$$M=\sup_{x \in [a,b]}f(x), \\ m = \inf_{x \in [a,b]}f(x). $$ 
Assuming $M > m$ (otherwise we have a trivial case of a constant function) we can choose
$$\delta < \frac{\epsilon}{6(M-m)},$$
and we then have,
$$U(P,f) - L(P,f) < \epsilon.$$
