Answer
Yes, the function $ f\left( x \right)=3x+4$ is continuous at $1$.
Work Step by Step
Consider the function $ f\left( x \right)=3x+4$,
First check whether the function is defined at the point $ a $ or not.
Find the value of $ f\left( x \right)$ at $ a=1$,
$\begin{align}
& f\left( 1 \right)=3\left( 1 \right)+4 \\
& =3+4 \\
& =7
\end{align}$
The function is defined at the point $ a=1$.
Now find the value of $\,\underset{x\to 1}{\mathop{\lim }}\,3x+4\,$,
$\begin{align}
& \,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right) \\
& =3\left( 1 \right)+4 \\
& =3+4 \\
& =7
\end{align}$
Thus, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7$
Now check whether $\,\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ or not.
From the above, $\,\underset{x\to 1}{\mathop{\lim }}\,\left( 3x+4 \right)\,=7\text{ and }f\left( 1 \right)=7$
Therefore, $\,\underset{x\to 1}{\mathop{\lim }}\,f\left( x \right)=f\left( 1 \right)$
Thus, the function satisfies all the properties of being continuous.
Hence, the function $ f\left( x \right)=3x+4$ is continuous at $1$.
