Discontinuity & its Types
Hey there! This time, we are discussing about the discontinuity and its various types viz. Jump Discontinuity, Essential Discontinuity, Removable & nonremovable 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:
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 “Nonremovable discontinuity”. This happens when there remains no relation in the gap of the graph. In case of function, Nonremovable 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 :
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…