What does discontinuity mean?
In terms of real life, discontinuity is equivalent to move up the pencil wen you plot a graph function. See below
With this idea in mind, there is several types of discontinuity.
Infinite jump discontinuity and
finite jump discontinuity
You can see this types in several internet pages. for example, this a finite jump discontinuity.
Mathematicaly, contnuity is equivalent to say that: