Continuity and discontinuity
Jump discontinuities
Here is an example of a discontinuous function. It has a jump discontinuity at . You can move and alter , the half-width of the interval in , in order to see how , the half-width of the interval in , changes. Try moving on top of a discontinuity, and see how the behaviour of the box changes.
Essential discontinuities
This is a different case of a discontinuous function: we have defined , but there is no limit as . This is called an essential discontinuity.
Other discontinuities and singularities
Other types of discontinuity exist, but these two illustrate the most common.
Something commonly thought of as a discontinuity, though in fact not, is the point in the function . In fact, is not defined at this point, and so the point is called a singularity. If we were to define a new function, equal to when and equal to when , then such a function would be defined at (obviously), but the function would still not be continuous at this point, as does not exist. It would be another example of an essential discontinuity.