Consider a function $f : R^2 \rightarrow R$ given by $f(x,y)= x^2 - y^3.$ I need to check whether the hypothesis of Implicit function theorem hold for $f$ at the point $(0,0)$.
For this, I first calculated the matrix $M(x,y)$ formed by differentiating this function with respect to $y$. $M(x,y) = -3y^2$. It's clearly non-invertible at $(0,0)$. I am stuck here as non-invertibility gives me an inconclusive test. How do I show that hypothesis of Implicit function theorem holds here or not?