Suppose $f:[0,1]\to\mathbf{R}$ is bounded and is continuous on each of the intervals $[0,u_1),(u_1,u_2),\ldots,(u_k,1]$ for some sequence of points $u_1<u_2<\cdots<u_k$ between 0 and 1. Then $f$ is integrable on $[0,1].$