Known as removable discontinuity or point discontinuity, this type of discontinuity occurs when a hole is present in the graph of a function. The National Science Digital Library (NSDL) states that when a point on the graph cannot fulfill the function a hole is present. The points on the graph that create graphing holes in removable discontinuities can often be plugged in and create a new continuous function.
An asymptotic discontinuity is a type of discontinuity where the graph of a function reaches a point on the graph but does not connect to it but continues after it, as defined by Brown University's Math Department. Asymptotic discontinuities can appear to have holes when graphed and can be seen on the graph both vertically and horizontally.
An infinite discontinuity occurs when a function is graphed and has an end that continues without ending. NSDL explains that this type of discontinuity stops can stop at one point on the graph, but the other end will continue on forever.
Also known as a simple discontinuity, a jump discontinuity occurs when a function is graphed and stops at one point on the graph and jumps to another. As explained by Brown University's Math Department, when a jump discontinuity is graphed it can have the same x-value but have two different y-values.