We all have heard the word “ continuity” while talking to someone or while reading something. But what if someone asks us the question, what is continuity in maths? The word Continuity comes from “continuous”. It means something that is endless or unbroken or uninterrupted. Therefore we can say that continuity is the presence of a complete path that we can trace on a graph without lifting the pencil. While it is ordinarily true that a continuous function has such graphs, but it won’t be a very precise or practical way to define what is continuity in maths. There are so many graphs and functions that are continuous or connected, in some places, while discontinuous, or broken, in other places. There are functions accommodating too many variables that are to be graphed by hand. Hence, it is extremely necessary that we have a more precise definition of what is continuity in maths. One that does not rely on our expertise to graph and trace a function.
Continuity Of A Function
The continuity of a function at a point can be defined in terms of limits. A function f(x) can be called continuous at x=a if the limit of f(x) as x approaching a is f(a). (A discontinuity can be explained as a point x=a where f is usually specified but is not equal to the limit. Sometimes singularities — points x=a where f is obscure — can also be counted as discontinuities.) A continuity of a function on an interval (or some other set) is continuous at each of the single points of that interval (or set). Usually, the term continuity of a function refers to a function that is basically continuous everywhere on its domain.
Conditions for Continuity
In calculus, a continuity of a function can be true at x = a, only if – all three of the conditions below are met:

The function is specified at x = a; i.e. f(a) is equal to a real number

The limit of the function as x addresses a exists

The limit of the function as x addressing a is equal to the function value at x = a
Solved Examples
Question 1) Is the function f(x) continuous at x = 0 in the graph below?
Solution 1) To check if the function is continuous at x = 0, we also have to check the three conditions:

First, we have to see if the function is defined at x = 0? Yes, f(0) = 2

Second, we have to see if the limit of the function f(x) as x approaches 0 exist? Yes

Lastly, we have to see if the limit of the function f(x) as x approaches 0 equal the function value at x = 0? Yes
Since all of the three conditions have met, we can say that f(x) is continuous at x = 0.
Example 2) Is f(x) continuous at x = 4 in the graph given below?
Solution 2) To examine for continuity at x = 4, we will have to check the same three conditions:

We have to see if the function is defined; f(4) = 2

Secondly, we have to check the limit exists

Lastly, the value of the function should not be equal to the limit; point discontinuity at x = 4