If $f : [a, b] → \mathbb{R}$ is a bounded function that $f(x) = 0$ for $x \not= c$, where $a < c < b$. Show that $f$ is Riemann integrable on $[a, b]$ and $\int_{a}^{b}f(x) dx = 0$.
I have a little confusion about this problem. I considered two cases $f(c)>0$ and $f(c)<0$ with partition $P_n$ as $P_n= \{a,x_1,x_2,...,b\} $ where $x_k=a+\frac{k(b-a)}{n}$ and assuming c $ \in [x_{k-1},x_{k} ]$ then I proved $\int_{a}^{b}f(x) dx = 0$ but my teacher considered two cases where c is inside the interval $ [x_{k-1},x_{k} ]$ and c is the one of the end point of $ [x_{k-1},x_{k} ]$.
Is that necessary?