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