Discontinuities are the points where a function fails to be continuous, and they come in three main types: removable, jump, and those caused by vertical asymptotes.
A removable discontinuity is a hole in the graph where the limit exists, but the function is not defined at that point, or it is defined differently.
Jump discontinuities occur when there is a sudden leap in function values, causing the left-hand and right-hand limits to differ.
Vertical asymptotes are where the function approaches infinity, and the limit does not exist.
The mechanism driving these discontinuities is the failure of the limit to match the function's value or to exist at all.
The cognitive trap here is confusing removable discontinuities with jump discontinuities.
They both involve limits, but removable ones have the same limit from both sides, while jumps do not.
Misidentifying these can lead to incorrect conclusions about a function's behavior, especially in problems requiring justification of continuity at a point.