Q. Let $\lambda$ denote Lebesgue measure on $\mathbf{R}$. Give an example of a continuous function $f:(0,1) \rightarrow \mathbf{R}$ such that $\lim _{n \rightarrow \infty} \int_{\left(\frac{1}{n}, 1\right)} f d \lambda$ exists (in $\left.\mathbf{R}\right)$ but $\int_{(0,1)} f d \lambda$ is not
defined.
I hope the area under $f$ from $1/n$ to $1$ need to be constant.
Any role for conditionally convergent series?
Best Answer
$f(x)=\dfrac{\sin(2\pi/x)}{x^2}$ is an example. Proof: Let $x=1/y$ to see
$$\tag 1\int_{[1/n,1]} \frac{\sin(2\pi/x)}{x^2}\,dx = \int_{[1,n]} \sin(2\pi y)\,dy.$$
Now the integral of $\sin(2\pi y)$ over any interval of length $1$ is $0.$ This implies the right side of $(1)$ is $0$ for any $n\in \mathbb N.$ But for $\int_0^1 f$ to exist as a real number, we must have $\int_0^1 |f|<\infty.$ You can check, using the same substitution $x=1/y,$ that $\int_0^1 |f|=\infty.$