One function with finitely many removable discontinuities is integrable. There are few pieces of evidence to proof this point.

First of all, let us think about a function is continuous and differentiable. The core of the integral is to find out the area under a function. It is obviously for us to think that there is a close area under the function in a certain interval. If we take a point away, the area would still stay same because one point would not affect the area. A few dots would not make the area of an image increase or decrease because a dot is too small so that we can ignore it.

Also, if we think the function, let’s say there exist infinite points, the infinite dots can form a certain area. Let’s imagine we paint lots of points on a paper, we will finally fill the whole paper. Similarly, if there are infinite dots, we could not determine the area under the function.

Therefore, in conclusion, one function with finitely many removable discontinuities is integrable.