To check this function is integrable or not, what we have to do is use the definition of integral. If we can prove this function is integrable, we can explain why the functions with finitely many removable discontinuities are integrable. The first thinf we have to do is partition the interval into n subintervals and select any sample point in the subintervals. If the function is integrable, we can find the area under the curve and it should exist and be equal in every sample points. In the question, it mentioned that there are “finitely” many removable discontinuities. It means the number of the discontinuities are not many as the number of irrational numbers. If there were infinitely many number of discontinuities are in the interval, the function might not be integrable. However, in this question, there are only finite number of the discontinuities. Therefore, the discontinuities do not really affect to the area under the curve.