Answer
The complete statement is, “A function f is continuous at a when three conditions are satisfied.
(a) f is defined at a, so that $ f\left( a \right)$ is a real number.
(b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists.
(c) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ $=$ $ f\left( a \right)$ ”
Work Step by Step
For a function to be continuous at a point a, the function must satisfy the following three conditions:
(a) f is defined at a.
(b) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ 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)=2x $
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)=2\left( 3 \right)=6$
The function is defined at the point $ a=3$.
Now find the value of $\,\underset{x\to 3}{\mathop{\lim }}\,2x $,
$\begin{align}
& \,\underset{x\to 3}{\mathop{\lim }}\,2x=2\underset{x\to 3}{\mathop{\lim }}\,x \\
& =2\left( 3 \right) \\
& =6
\end{align}$
Thus, $\,\underset{x\to 3}{\mathop{\lim }}\,2x=6$
From the above two steps, $\,\underset{x\to 3}{\mathop{\lim }}\,2x=6=f\left( 3 \right)$.
Thus, the function satisfies all the conditions of being continuous.
Hence, the function $ f\left( x \right)=2x $ is continuous at $3$.
