Discontinuity & its Types

Hey there! This time, we are discussing about the discontinuity and its various types viz.- Jump Discontinuity, Essential  Discontinuity, Removable & non-removable discontinuity. First of all, we must know what is discontinuity and how to get idea whether a function is discontinuous..

What is Discontinuity-

Discontinuity in initial stage, we explain using graph.-

Consider a graph of some function, the graph is smooth initially, but at some point, due to some change in value or sign, our graph takes a bit or long gap in its continuity, this gap or break in continuity is the ‘discontinuity’.

above figure explains the site of various discontinuities in graphical format:

discontinuity in graph
Discontinuity in graph

Types of Discontinuity:

If a function f is not continuous at some point x, then f is discontinuous at x.
There are different cases of discontinuity:

  • f(x).f(x),f(x+)all exist but are not all equal.

This is a discontinuity of first kind or “Jump discontinuity”. This happens when there is a long gap in the graph. In case of function, Jump discontinuity happens when f(x), f(-x) & f(+x) exist, but all are unequal.

  • f(x+) or f(x-) do not exist.

This is a discontinuity of second kind or “Essential discontinuity”. This happens when there is a break in the graph. In case of function, Essential discontinuity happens when f(x), f(-x) & f(+x) do not exist.

  • f(x+) ≠ f(x-), f(x) is immaterial.

This is Ordinary discontinuity  or “Non-removable discontinuity”. This happens when there remains no relation in the gap of the graph. In case of function, Non-removable discontinuity happens when f(-x) & f(+x) exist, but all are unequal, here, f(x) is immaterial.

  • f(x+) = f(x-) ≠ f(x).

This is a Removable discontinuity. This happens when there is a difference in the point in the graph & the calculated point. In case of function, Removable discontinuity happens when f(x)≠ f(-x) = f(+x). This can be removed by redefining f(x) so that f(x)=f(-x) = f(+x).

Figure below explains types of discontinuity  (Jump discontinuity, Essential  Discontinuity, Removable & non – removable discontinuity) in brief :-

Types of discontinuity
discontinuity & its types

We’ve discussed discontinuity in this post. Get the knowledge of limit & its theorems in our previous posts. If you have any doubts about this section, feel free to ask in comment section below…