In order to judge whether a function is integrable or not, firstly it must be defined for each points on the graph. As the question stated, the functions with finitely many removable discontinuities are defined everywhere. Then let the functions are partitioned into n subintervals. Consider the area in each subinterval as a regular rectangle with the width of each subintervals. Then choose a sample point for each subinterval as the length of the rectangle. If it is integrable, whatever the sample is chosen, the area should always exist and equal. For a function with finitely many removable discontinuities, when the continuous parts are chosen or the discontinuous parts are chosen, one overestimate the area, the other underestimate the area. If the results are same in these two situation, the integral should be same as the results. If the discontinuities are finite, when it multiply with the width which is extremely small, the result will be negligible. The main part are always the infinite many continuities times the length. Finitely many removable discontinuities is not powerful enough to change the answer, cause as the n increase, the width is extremely small. Functions with finitely many removable discontinuities are integrable.