# Limit of an integral over a ball as the radius of the ball goes to zero 2

Let $$D(a,r)$$ be an open ball in $$\mathbb{R}^{k}$$ ($$k\geq1$$), and $$f$$ locally integrable function in $$\mathbb{R}^{k}$$. Do we have: $$\lim_{r\to 0}\int_{D(a,r)}f(t)dt=0?$$

• Yes, by dominated convergence. – Angina Seng Oct 26 '18 at 16:45
• In physics there are cases where one uses point charges to describe fields. A point charge is represented by the Dirac delta function. If you integrate over a sphere around the point charge, and then shrink the radius to $+0$, what you get is a constant term, not necessarily zero. – M. Wind Oct 26 '18 at 18:02
• The question says explicitly "$f$ is a locally integrable function" ruling out the possibility that $f$ is a Dirac's delta. – Julián Aguirre Oct 26 '18 at 18:21
• Lord Shark the Unknown: How would you deduce this from dominated convergence? I don't see it. – M. Rahmat Oct 27 '18 at 4:54
• Why would it not be 0? Out of all the numbers out there, such as: 1, 15, 0, 92, 5673, -14i, why would it be anything other than 0 ? – Nike Dattani Oct 28 '18 at 22:08