Given the following function: $$ f(x,y)=\frac{\sin xy}{e^x-y^2}\;, $$ is the following justification correct?
As sin is a continuous function, the exponential is a continuous function, the polynomials are continuous functions and the subtraction of continuous functions are continuous functions, both the numerator and denominator are continuous functions, hence the quotient of two continuous functions is a continuous function for all values of its domain.
The ratio of 2 continuous functions is continuous in points where denominator is not null. So, in your case, you need $e^x \ne y^2$.