Riemann integrability of indicator function of compact subset of a closed interval

Riemann integrability of indicator function of compact subset of a closed
interval

Let $K\subset[0,1]$ be compact and consider the function $1_K:$ $$
1_K(x)=\begin{cases} 0 & \text{if } x \not\in K \\ 1 & \text{if } x \in K
\end{cases} $$
My question: is $1_K$ Riemann integrable?
According to the Lebesgue's criterion for Riemann integrability, it
suffices to know that if $m(A)=0$ where $m$ is the Lebesgue measure, and
$A$ is the set of points of discontinuity of $1_K$.
The question is trivial when $K$ is a finite union of closed intervals.
But I don't see how to deal with the general case.