Showing Inverse Doesn't Exist Locally

Question

I'm trying to show that the function $$F(u,v) = [5u^2-v^4, 3uv]$$ is not invertible locally at (0,0) which is true because the Jacobian at that point is in fact 0, but regardless of the Inverse Function Theorem failing, I need to prove the inverse cannot exist at that point. I'm not sure how to explicitly show this because it seems relatively trivial given the statement of the theorem

Answer 1

As already answered by @saulspatz in the comments:

Given: $$F(u,v) = [5u^2-v^4, 3uv] $$

One notices: $$ F(-u,-v) = [5(-u)^2-(-v)^4, 3(-u)(-v)]= [5u^2-v^4, 3uv] $$

Hence for any non-zero $u_1,v_1$ we can set $u_2=-u_1$ and $v_2=-v_1$ such that:

$$ F(u_1,v_1)=F(u_2,v_2) \,\,\, \wedge \,\,\, (u_1,v_1)\neq (u_2,v_2) $$

Hence $F$ is not injective, and therefore cannot be invertible.