Question

Let M be a positive real number and let u, v: $\R^2\rightarrow\R$ be continuous functions satisfying $\sqrt{u(x,y)^2+v(x,y)^2}\ge M\sqrt{x^2+y^2}\ for\ all\ (x,y)\isin\R^2.$
Let F: $\R^2\rightarrow\R^2$ be given by
F(x, y) = (u(x, y), v(x, y)) for (x, y)$\isin\R^2$.
Then which of the following is/are true?

• A. F is injective.
• B. If K is open in $\R^2$, then F(K) is open in $\R^2$.
• C. If K is closed in $\R^2$, then F(K) is closed in $\R^2$.
• D. If E is closed and bounded in $\R^2$, then F-1(E) is closed and bounded in $\R^2$.

Answer 1

Answer: (C), (D)

If K is closed in $\R^2$, then F(K) is closed in $\R^2$.

Answer 2

The correct option is (C): If K is closed in $\R^2$, then F(K) is closed in $\R^2$. and (D): If E is closed and bounded in $\R^2$, then F-1(E) is closed and bounded in $\R^2$.