Here is my question that I am struggling with:
Assume $a, b \in \mathbb R, a \lt b, c \in [a, b], d \in \mathbb R, d \neq 0$. Consider $f : [a, b] \to \mathbb R$ defined by $ f(x) = \begin{cases} 0, &x \in [a, b] \smallsetminus \{c \}, \\ d, &x = c. \end{cases} $ Use the definition of the Riemann integral to show that $\int_{a}^{b} f = 0$.
I know the definition of a Riemann integral has $\|P\| \lt \delta$, $|S(f:\dot P)-L|<\epsilon$, and that $S(f:\dot P) = \sum f(t_{i})(x_{i}-x_{i-1})$, but I'm not sure how to use that to prove that the integral is $0$. Any tips?
Added based on the comments. $P$ is a uniform partition, $P= \{x_i \}_{i=0}^n$, and $\dot P$ is a tagged partition, $\dot P= \{ t_i \}_{i=1}^{n}$.