Answer
The statement “If $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\ne f\left( a \right)$ and $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists. I can redefine $ f\left( a \right)$ to make f continuous at a.” makes sense.
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) $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$
Since, $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)\ne f\left( a \right)$ and $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$ exists.
So the function does not satisfy the third condition of being continuous at a.
If the value of $ f\left( a \right)$ is redefined and is taken equal to $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)$, then, $\underset{x\to a}{\mathop{\lim }}\,f\left( x \right)=f\left( a \right)$ and the function satisfies all the conditions of being continuous
Then the function is continuous at a.
Hence, the statement makes sense.
