Answer
The statement “f and g are both continuous at $ a $, although $\frac{f}{g}$ is not” 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, f and g are continuous functions at a, both of them satisfy the above three conditions.
Now check whether the function $\frac{f}{g}$ is continuous at a, or not.
Find the value of $\left( \frac{f}{g} \right)\left( x \right)$ at $ a $,
$\left( \frac{f}{g} \right)\left( a \right)=\frac{f\left( a \right)}{g\left( a \right)}$
If $ g\left( a \right)=0$, then the value $\left( \frac{f}{g} \right)\left( a \right)$ is not defined.
In this case, the function $\left( \frac{f}{g} \right)\left( x \right)$ does not satisfy all the conditions of being continuous.
Thus, the function $\left( \frac{f}{g} \right)\left( x \right)$ is not continuous at a.
Hence, the statement makes sense.
