I am having difficulties proving the equivalence of these statements.

Show that the following statements are equivalent

 

(1) $f: D \to \mathbb R^n$ is continuous.

 

(2) For every closed ball $B$ in $\mathbb R^n$ , the inverse image of $B$ under $f$ is closed in $D$.

 

(3) For every closed subset $S$ of $\mathbb R^n$, the inverse image of $S$ under $f$ is closed in $D$.

So to show equivalence I have to show the following implication chain right? $(1)\Rightarrow (2) \Rightarrow (3)\Rightarrow (1)$

I do understand the analogous proof for open balls and subsets but I don't know where to start based from this..I would appreciate any help!

Thank you

Maria

I am having difficulties proving the equivalence of these statements.

Show that the following statements are equivalent

(1) $f: D \to \mathbb R^n$ is continuous.

(2) For every closed ball $B$ in $\mathbb R^n$ , the inverse image of $B$ under $f$ is closed in $D$.

(3) For every closed subset $S$ of $\mathbb R^n$, the inverse image of $S$ under $f$ is closed in $D$.

So to show equivalence I have to show the following implication chain right? $(1)\Rightarrow (2) \Rightarrow (3)\Rightarrow (1)$

I do understand the analogous proof for open balls and subsets but I don't know where to start based from this..I would appreciate any help!

Thank you

Maria

I am having difficulties proving the equivalence of these statements.

Show that the following statements are equivalent

(1) f: D -> R^n is continuous.

(2) For every closed ball B in R^n , the inverse image of B under f is closed in D.

(3) For every closed subset S of R^n, the inverse image of S under f is closed in D.

So to show equivalence I have to show the following implication chain right? (1)=>(2)=>(3)=>(1)

I do understand the analogous proof for open balls and subsets but I don't know where to start based from this..I would appreciate any help!

Thank you

Maria

I am having difficulties proving the equivalence of these statements.

Show that the following statements are equivalent

(1) $f: D \to \mathbb R^n$ is continuous.

(2) For every closed ball $B$ in $\mathbb R^n$ , the inverse image of $B$ under $f$ is closed in $D$.

(3) For every closed subset $S$ of $\mathbb R^n$, the inverse image of $S$ under $f$ is closed in $D$.

So to show equivalence I have to show the following implication chain right? $(1)\Rightarrow (2) \Rightarrow (3)\Rightarrow (1)$

I do understand the analogous proof for open balls and subsets but I don't know where to start based from this..I would appreciate any help!

Thank you

Maria