# Rudin 6.7

1. May 1, 2008

### ehrenfest

1. The problem statement, all variables and given/known data
Suppose f is a real function on (0,1] and f is Riemann-integrable on [c,1] for every c>0. Define

$$\int_0^1 f(x) dx = \lim_{c\to 0} \int_c^1 f(x) dx$$

if this limit exists and is finite.

Construct a function f such that the above limit exists, although it fails to exist with |f| in place of f.
2. Relevant equations

3. The attempt at a solution
I think f(x) = sin(1/x)/x will work although I am having trouble proving it. It seems intuitively true that the limit exists since the positive and negative humps will cancel and the oscillations in the integral will get get smaller and smaller as c gets smaller and smaller. I am not really sure how to prove that the limit diverges when we take the absolute value of f.