Question

Why do we wanna find points of discontinuity？

Answer 1

When you are asked to study a function, you will need to know if a function is or is not continuous.
When it is not, you'll have to see what happens in these points, and study the behaviour using limits.
Then you see what kind of discontinuity it is.

It's all part of the function study.

Answer 2

just to check where the functions is not defined