Give an example of a function having infinite number of point of discontinuities but integrable. Justify.

Question

Give an example of a function of one variable having infinite number of point of discontinuities but integrable. Justify.

Please help me.

Answer 1

The classical example is Thomae's function. It is discontinuous at every rational. On the other hand, if $P$ is a partition of $[0,1]$, then $L(f,P)=0$. It's not hard to prove that$$\inf\left\{U(f,P)\,\middle|\,P\text{ partition of }[0,1]\right\}=0.$$

Answer 2

Let $$I_n = \left[\frac 1n, \frac{1}{n+1}\right]$$ and $$ f = \sum\limits_{n=1}^{+\infty} (-1)^n \ \bf 1_{I_n}$$ then $f$ is integrable since the alternatic harmonic series converges.