Example of a Continuous Function

Let’s take an example to find the continuity of a function at any given point.

Consider the function of the form

f\left( x \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{{x^2} - 16}}{{x - 4}},\,\,\,if\,x \ne 4} \\ {0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,if\,x = 4} \end{array}} \right.

We shall check the continuity of the given function at the point x = 4.

To check the continuity of the given function we follow the three steps.

(i) Value of the Function at the Given Point
We have given value of function at x = 4 is equal to 0.

f\left( 4 \right) = 0

(ii) Limit of the Function at the Given Point

\begin{gathered} \mathop {\lim }\limits_{x \to 4} f\left( x \right) = \mathop {\lim }\limits_{x \to 4} \frac{{{x^2} - 16}}{{x - 4}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 4} f\left( x \right) = \mathop {\lim }\limits_{x \to 4} \frac{{{{\left( x \right)}^2} - {{\left( 4 \right)}^2}}}{{x - 4}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 4} f\left( x \right) = \mathop {\lim }\limits_{x \to 4} \frac{{\left( {x + 4} \right)\left( {x - 4} \right)}}{{x - 4}} \\ \Rightarrow \mathop {\lim }\limits_{x \to 4} f\left( x \right) = \mathop {\lim }\limits_{x \to 4} \left( {x + 4} \right) \\ \end{gathered}

Applying the limits, we have

\Rightarrow \mathop {\lim }\limits_{x \to 4} f\left( x \right) = 4 + 4 = 8

(iii) From the above information it is clear that

\mathop {\lim }\limits_{x \to 4} f\left( x \right) \ne f\left( 4 \right)

Hence the function f discontinues at the point x = 4.