# Continuous function with 2 discontinuous points is Riemann integrable

Suppose that $$g:[a,b]\to\mathbb{R}$$ is continuous except at $$r_1,r_2\in(a,b)$$. Prove that $$g$$ is Riemann integrable on $$[a,b]$$.

Intuitively, I know this is true because the upper sum and lower sum will only differ by the contribution from $$f(r_1)$$ and $$f(r_2)$$, and you refine the partition until it disappears. But I'm not quite sure how to turn that into a formal proof.

• MAybe you can do it in a similar way that I do in this answer: math.stackexchange.com/questions/2980279/rieman-integration/… – Tito Eliatron Nov 2 '18 at 20:07
• You also need to assume that $g$ is bounded on $[a, b]$. The result can be easily generalized: if the function $g:[a, b]\to\mathbb {R}$ is bounded on $[a, b]$ and if the set $D$ of its discontinuities on $[a, b]$ has a finite number of limit points then $g$ is Riemann integrable on $[a, b]$. – Paramanand Singh Nov 3 '18 at 2:46
• The crux of the argument is to show that a single discontinuity does not matter and without loss of generality one can take this single discontinuity at end point of interval of integration. Start by proving that if $g$ is bounded on $[a, b]$ and continuous on $(a, b]$ then $f$ is Riemann integrable on $[a, b]$. – Paramanand Singh Nov 3 '18 at 2:51