== lim f(x) when x --> x_o exists iff both one-sided limits exist and they're identical
== x = a is a removable discontinuity of function f if f(a) exists and also limf(x) when x --> a exists and finite, but they aren't equal
I'm not sure what you guys call "an infinite discontinuity" to, but if it refers to a point a s.t. lim f(x) when x --> a from the right or from the left (or from both) is oo or -oo, then...well, calculate the limits.
== jump discontinuity is when both one-sided limits exist finitely both they're different.
Now do some maths