Is it possible to find the area under a curve with an infinite number of holes (removable discontinuities)?

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After learning about improper integrals, we discussed about the idea that a hole (removable discontinuity) has no effect on the area under a curve since it is just a single point and therefore has no width and it is pretty fascinating.

However, I then have to ask, what if there were an infinite number of holes along the function (for example, there is a hole every multiple of 5 or something) could you still find an area? As a side question to that, what if the whole function consisted of holes (is that even possible?) and if so could you find the area under it?