Suppose $f:[a,b] \rightarrow \mathbb{R}$ is bounded and continuous at all points except $c$ where $a < c <b$. Prove that $f$ is Riemann integrable.

Suppose $f:[a,b] \rightarrow \mathbb{R}$ is bounded and continuous except at a single point $c\in(a,b)$. Prove that $f$ is Riemann integrable.

The way I want to go about this is to split the domain and say that there are 2 partitions. But I'm unsure what to do with the upper and lower Darboux sums, since they belong to 2 different parts of the domain.

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1 Answer

$\bullet$ enclose c in $P_0 = [c-\frac{\delta_1}{2},c+\frac{\delta_1}{2}]$ and $|f(x)| < k$ $\forall x \in [a,b]$

$\bullet$ consider partition $P_1$ for $[a,c-\frac{\delta_1}{2}]$ and $P_2$ for $[c+\frac{\delta_1}{2},b]$ with $\delta_1 < \frac{\epsilon}{4k}$

on $[c-\frac{\delta_1}{2},c+\frac{\delta_1}{2}]$ :$M^1$ = $sup_{x \in [c-\frac{\delta_1}{2},c+\frac{\delta_1}{2}]} f(x)$ and $m^1$ = $inf_{x \in [c-\frac{\delta_1}{2},c+\frac{\delta_1}{2}]} f(x)$

then $M^1 - m^1 <2k$

for $P_1$ and $P_2$ we have $U(P_i,f) - L(P_i,f) < \frac{\epsilon}{4}$ where $i = 1$ and $2$

then consider a new partition $P$ of $[a,b]$ and where $P =P_1 \cup P_2 \cup P_0$ then

$U(P,f)-L(P,f) < [U(P_1,f) - L(P_1,f)] + [U(P_1,f) - L(P_2,f)] + [M^1 - m^1]\delta_1$

$< 2 \cdot \frac{\epsilon}{4} + 2k\cdot \frac{\epsilon}{4k}$

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