How do I determine the discontinuity of a function?

Hint: These types of problems are based on the concept of factorising functions and then find the discontinuity points. We first have to factorise the given function in such a way that the denominator with a variable cancels out, and then the function is continuous. If the function has a variable in the denominator, then the point for which the denominator alone becomes 0 is the discontinuity point and thus the function is not continuous. In such a manner we can find whether the function is continuous or not.

Complete step by step answer:
According to the question, we are asked how to determine whether a function is continuous or not.
We can explain the concept with the help of an example.
Let us consider a function \[\dfrac{{{x}^{2}}-{{a}^{2}}}{x+a}\].
Here, we find that on substituting the value of x as –a, we get 0 in the denominator which means the function is not defined at the point –a.
Therefore, the function is discontinuous at the point –a.
But it is wrong.
We first have to factorise the function in such a way that the denominator is constant.
Here, we know that \[{{x}^{2}}-{{a}^{2}}=\left( x+a \right)\left( x-a \right)\]. Substituting this in the assumed function, we get
\[\dfrac{{{x}^{2}}-{{a}^{2}}}{x+a}=\dfrac{\left( x+a \right)\left( x-a \right)}{x+a}\]
Cancelling out the common terms from the numerator and denominator, we get
\[\dfrac{{{x}^{2}}-{{a}^{2}}}{x+a}=x-a\]
Here, the denominator is 1 and we find that there is no point in the function where it is not determined.
Therefore, the function \[\dfrac{{{x}^{2}}-{{a}^{2}}}{x+a}\] is continuous function and there is no discontinuity point.
Let us now consider a function \[\dfrac{{{x}^{2}}+{{a}^{2}}}{x+a}\].
Here, we cannot further simplify the numerator in such a way that we get a common term (x+a).
Hence we cannot further factorize it.
Now, we find that, when x=-a, the denominator is 0, which means the function is not determined at the point x=-a.
Therefore, the assumed function \[\dfrac{{{x}^{2}}+{{a}^{2}}}{x+a}\] is discontinuous at the point x=-a.
Hence, the point of discontinuity for the function \[\dfrac{{{x}^{2}}+{{a}^{2}}}{x+a}\] is x=-a.

Note:
We can also try to find the points of discontinuity by assuming different functions. Do not forget to factorise the function before stating the discontinuity. We should consider the signs during factorisation and also mind the sign conventions. We can also plot graphs for the functions to check whether the function is continuous or not.