If a function is defined as
where both are continuous functions, then
is also continuous where defined, i.e. where
So, in your case, this function is continous everywhere, except where
To solve this equation, we can use the formula
It means that, if the leading terms is 1, then the x coefficient is the opposite of the sum of the roots, and the constant term is the product of the roots.
So, we're looking for two terms whose sum is 7, and whose product is 12. These numbers are easily found to be 3 and 4.
So, this function is continuous for every real number different than 3 or 4.