Answer
The given function is discontinuous at $ x=3$.
Work Step by Step
Consider the given function.
First check the discontinuity of the function at $ x=0$.
Find the value of $ f\left( x \right)$ at $ x=0$,
From the definition of the function, for $ x=0$, $ f\left( x \right)=x+7$
Then the value of $ f\left( x \right)$ at $ x=0$ is,
$ f\left( 0 \right)=0+7=7$
The function is defined at the point $ x=0$.
Now find the value of $\,\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)$,
First find the left-hand limit of $\,f\left( x \right)$,
That is,
$\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f\left( x \right)=0+7=7$
Now find the right-hand limit of $\,f\left( x \right)$,
That is,
$\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f\left( x \right)=7$
Since the left-hand limit and right-hand limit are equal, that is $\underset{x\to {{0}^{-}}}{\mathop{\lim }}\,f\left( x \right)=7=\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,f\left( x \right)$.
Thus $\,\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=7$.
From the above steps, $\,\underset{x\to 0}{\mathop{\lim }}\,f\left( x \right)=7=f\left( 0 \right)$.
Thus, the function satisfies all the properties of being continuous.
Thus, the given function is continuous at $ x=0$.
Now check the discontinuity of the function at $ x=3$
Find the value of $ f\left( x \right)$ at $ x=3$,
From the definition of the function,
$ f\left( 3 \right)=7$
The function is defined at the point $ x=3$.
Now find the value of $\,\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$,
First find the left-hand limit of $\,f\left( x \right)$,
That is,
$\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)=7$
Now find the right-hand limit of $\,f\left( x \right)$,
That is,
$\underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)={{3}^{2}}-1=8$
Since the left-hand limit and right-hand limit are not equal, that is $\underset{x\to {{3}^{-}}}{\mathop{\lim }}\,f\left( x \right)\ne \underset{x\to {{3}^{+}}}{\mathop{\lim }}\,f\left( x \right)$
Thus, $\,\underset{x\to 3}{\mathop{\lim }}\,f\left( x \right)$ does not exist.
Thus, the function does not satisfy the second property of being continuous. Hence, the given function is discontinuous at $ x=3$.
