Riemann Integrable Function

Let $H(x)=k$ for $x =\frac{1}{k} \ (k \in \mathbb{N})$ and $H(x)=0$ elsewhere on $[0,1]$ . Is $H$ riemann integrable?

Mistaken post. Sorry.

Quote:

Originally Posted by Chandru1

Let $H(x)=k$ for $x =\frac{1}{k} \ (k \in \mathbb{N})$ and $H(x)=0$ elsewhere on $[0,1]$ . Is $H$ riemann integrable

Suppose that $T = \left\{ {\frac{1}{k}:k \in \mathbb{Z}^ + } \right\}$ . That is the set of discontinuities.
$T$ is countable so the function is Riemann integrable.

Here is an outline of a proof. In any partition of $[0,1]$ the subinterval that contains 0 contains almost all points of $T$ . The points outside of that can be covered by intervals the sum of their is as small as needed. Thus showing the integral is zero.