1. function integrable

Suppose $\displaystyle a<c<b<d$. Let $\displaystyle k$ be any non-zero real number. Define $\displaystyle f: \mathbb{R} \to \mathbb{R}$ as follows: $\displaystyle f(x) = \begin{cases} 0 \ \ \ \ \ \text{if} \ x < c \\ k \ \ \ \ \ \text{if} \ c \leq x \leq d \\ 0 \ \ \ \ \ \text{if} \ d < x \end{cases}$

(a) Prove that $\displaystyle f$ is Riemann Integrable on $\displaystyle [a,b]$. What is its integral?

(b) If every $\displaystyle <$ in definition of $\displaystyle f$ were changed to $\displaystyle \leq$ and vice versa, how would the answer change?

For (a) the integral would be $\displaystyle k(d-c)$. Here is a proof. Proof. Let $\displaystyle \epsilon >0$. Choose $\displaystyle \delta >0$ such that $\displaystyle \delta < \frac{\epsilon}{2|d-c|}$. Let $\displaystyle P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n-1}, x_{n} \}$ be a partition of $\displaystyle [a,b]$ with $\displaystyle ||P|| < \delta$. Choose $\displaystyle x_{1}^{*}, x_{2}^{*}, \ldots , x_{n}^{*}$ arbitrarily such that $\displaystyle x_{i-1} \leq x_{i}^{*} \leq x_{i}$. Then $\displaystyle \mathcal{R}(f,P) = \sum_{i=1}^{n} f(x_{i}^{*})(x_{i}-x_{i-1})$. We have to break this up into three sums? Because ultimately we want to show that $\displaystyle |\mathcal{R}(f,P)-k(d-c)| < \epsilon$.

2. For any such partition, you can make a finer partition by adding c and d as "break points". Once you do that, the three sums are simple.

3. Originally Posted by HallsofIvy
For any such partition, you can make a finer partition by adding c and d as "break points". Once you do that, the three sums are simple.
Because then two sums are $\displaystyle 0$ and we are only worried about the "middle" sum?