Given that $g(x,y)=x^3y+xy^3$, suppose we wish to compute $$\int_{C}{\nabla g\cdot\textrm{d}\mathbf{r}},$$ where $C$ is the contour line $g=5$. The solution given is that since the gradient is always orthogonal to contour lines, the line integral evaluates to $0$.
However, the graph of $C$ has two asymptotes, namely at $x=0$ and $y=0$. So what does it mean to line integrate over $C$ in this case? Do we only integrate wherever $C$ is continuous?
How about in general when $C$ has a removable discontinuity instead of an asymptote?