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The examples here cover most commonly encountered types of  discontinuity; another is exhibited by the square root function. It is undefined for x < 0, so is not continuous at x = 0.

One can also have functions that have a finite limit on one side, and go to infinity on the other.

Finally there are functions that  are discontinuous over a range. (One curious if artificial example: the function that is 1 on rational numbers and 0 on irrational real numbers.)