Show that the domain is closed

I have to show that the domain defined by the half unit circle: $\left \{ \left ( x,y \right )\in \mathbb{R}^2: x^2+y^2\leq 1, x\geq 0 \right \}$

for the function $f:D\rightarrow \mathbb{R}$ is closed (the function is $f(x,y)=2xy^2-2x^2$), so that I can argue the function has a min/max value. I have shown it is bounded and continuous.

Solution 1:

Hint : You can write$$\left \{ \left ( x,y \right )\in \mathbb{R}^2: x^2+y^2\leq 1, x\geq 0 \right \} = g^{-1}([0,1]\times [0, +\infty))$$

where $g : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ is the continuous function defined for all $(x,y) \in \mathbb{R}^2$ by $$g(x,y)=(x^2+y^2,x)$$