Is this function (discontinuous at one point) integrable?

Question

Consider the function $f:[0,1] \to \mathbb{R}$ defined by $f(0)=0$ and $$ f(x)=2x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right). $$ Is $f$ integrable on $[0,1]$?

Ps. Clearly $f$ is discontinuous at $0$ and it has primitive $F(x)=x^2 \sin\left(\frac{1}{x}\right)$. But I don't know whether $\int_0^1 f(x)dx=F(1)-F(0)=\sin 1$. Also, I would say that $f$ is integrable if and only if $g(x)=\cos(1/x)$ for $x>0$ and $g(0)=0$ is integrable on $[0,1]$ (since the remaining term is continuous on $[0,1]$).

score 12 · Accepted Answer · answered Apr 08 '18 at 22:24

12

Yes, it is integrable, since it is bounded and the set of the points at which it is discontinuous has measure $0$.

answered Apr 08 '18 at 22:24

José Carlos Santos

427,504

We can apply monotone convergence theorem to find the value of the integral, right? – Apr 08 '18 at 22:31
Yes. And you can also do$$\int_0^1f(x),\mathrm dx=\lim_{a\to0}\int_a^1f(x),\mathrm dx=\lim_{a\to0}F(1)-F(a)=F(1)-F(0),$$since $F$ is continuous. – José Carlos Santos Apr 08 '18 at 22:34
1

@JoséCarlosSantos For the monotone convergence theorem, would you make the sequence $f_n(x) = f(x) \cdot X_{[\frac 1n, 1]} - X_{[0, \frac 1n]}$? (since $f \ge -1$ on $[0, 1])$. – Ovi Apr 09 '18 at 01:38
@Ovi Yes, that would do. – José Carlos Santos Apr 09 '18 at 06:20

score 2 · Answer 2 · answered Apr 08 '18 at 22:35

2

Because $F'(x)=2x\sin(1/x)-\cos(1/x)$ for $x\ne 0$ and $F'(0)=0$ and that $\displaystyle\int_{0}^{1}|F'(x)|dx\leq 3<\infty$, $F$ is actually absolutely continuous, so $\displaystyle\int_{0}^{1}f(x)dx=F(1)-F(0)=\sin 1$.

answered Apr 08 '18 at 22:35

user284331

55,591

score 1 · Answer 3 · answered Apr 09 '18 at 02:21

Note that the function $f$ in your question is bounded on any finite interval and is discontinuous only at a single point hence it is Riemann integrable on any closed interval.

And clearly you have found a primitive $F$ such that $F'=f$ everywhere thus by second fundamental theorem of calculus $$\int_{a} ^{b} f(x) \, dx=F(b) - F(a) $$

Is this function (discontinuous at one point) integrable?

3 Answers3