Question

If a function f:[ a,b] to R has finitely many points of discontinuity in [ a,b] , then f is integrable on [ a,b].

True or false with full explanation

Answer 1

Solution:

True.

Proof:

Suppose $c \in(a, b)$ . Let $\varepsilon>0$ . Since f is bounded, there exists M>0 such that $|f(x)| \leq M$ for all $x \in[a, b]$ . Let $\delta=\varepsilon / 12 M$ . Since f is continuous on$[a, c-\delta]$ and $[c+\delta, b]$ , f is Riemann-integrable on $[a, c-\delta]$ and $[c+\delta, b]$ , so there exists partitions $P_{1}=\left\{a=x_{0}<\cdots<x_{n}=c-\delta\right\} of [a, c-\delta]$ and $P_{2}=\left\{c+\delta=y_{0}<\cdots<y_{m}=c-\delta\right\}$ of $[c+\delta, b]$ such that

$U\left(P_{1}, f\right)-L\left(P_{1}, f\right)<\frac{\varepsilon}{3} \quad \text { and } \quad U\left(P_{2}, f\right)-L\left(P_{2}, f\right)<\frac{\varepsilon}{3}$

Consider the partition of [a, b] given by $P=P_{1} \cup P_{2}$ . Then

$\begin{aligned} U(P, f) &=\sum_{i=1}^{n} M_{i} \Delta x_{i}+2 \delta \sup _{x \in[c-\delta, c+\delta]} f(x)+\sum_{j=1}^{m} M_{j} \Delta y_{j} \leq U\left(P_{1}, f\right)+2 M \delta+U\left(P_{2}, f\right) \\ L(P, f) &=\sum_{i=1}^{n} m_{i} \Delta x_{i}+2 \delta \inf _{x \in[c-\delta, c+\delta]} f(x)+\sum_{j=1}^{m} m_{j} \Delta y_{j} \geq L\left(P_{1}, f\right)-2 M \delta+L\left(P_{2}, f\right) \end{aligned}$

and so

$\begin{aligned} U(P, f)-L(P, f) & \leq\left[U\left(P_{1}, f\right)-L\left(P_{1}, f\right)\right]+4 M \delta+\left[U\left(P_{2}, f\right)-L\left(P_{2}, f\right)\right] \\ &<\frac{\varepsilon}{3}+4 M \times \frac{\varepsilon}{12 M}+\frac{\varepsilon}{3}=\varepsilon \end{aligned}$

Hence f is Riemann integrable.