If we are looking for the integral of a function, we find the area under the function within of an interval. So we will divide the graph into many small rectangles and add up all the areas of the rectangles. When there is a function that has many removable discontinuities but finite, it means that there is a limit of numbers that has a different area of a rectangle. So as we divide the function into a huge number of subintervals n, say 1000, the width of each rectangle will be relatively small. Therefore, the areas of these finite points of removable discontinuities will be very small.

In conclusion, as n we choose is very big, 1/n, which is the width of each rectangle, will be infinitely small, and it will make the areas of these rectangles infinitely small. Then they can be negligible. Thus, the area under the graph of the function will be the total of the rectangles that are made of the continuous points. So that’s why a function with finite many removable discontinuities can still be integrable.