Answer
A function f is continuous at a number a, when it satisfies the following conditions:
(a) The function f is defined at a.
(b) The value of $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists.
(c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$
Work Step by Step
A function f is continuous at a number a, if it satisfies the following conditions:
(a) The function f is defined at a.
(b) The value of $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists.
(c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$
For example, consider a function $ f\left( x \right)=x+2$
To check whether the function is continuous at the point $ a=3$ or not,
find the value of $ f\left( x \right)$ at $ a=3$,
$ f\left( 3 \right)=3+2=5$
The function is defined at the point $ a=3$.
Now find the value of $\,\underset{x\to 3}{\mathop{\lim }}\,x+2$,
$\begin{align}
& \,\underset{x\to 3}{\mathop{\lim }}\,x+2=\underset{x\to 3}{\mathop{\lim }}\,x+2 \\
& =3+2 \\
& =5
\end{align}$
Thus, $\,\underset{x\to 3}{\mathop{\lim }}\,x+2=5$
From the above two steps, $\,\underset{x\to 3}{\mathop{\lim }}\,x+2=5=f\left( 3 \right)$
Thus, the function satisfies all the conditions of being continuous.
Hence, the function $ f\left( x \right)=x+2$ is continuous at $3$.
