A point x=a is called a point/removable discontinuity if the limit of f(x) at x=a exists and is equal to L (as a two-sided limit), and f(a) is defined, but f(a) is not equal to L. A point x=a is called a removable singularity if the limit of f(x) at x=a exists, but f(a) is undefined. These are called point/removable discontinuities/singularities because you could make f continuous (at a) just by modifying f at a single point, x=a.