Homework Statement

If f is a continuous mapping of a metric space X into a metric space Y, Let E be any subset of X. How to show, by an example, that f($$\overline{E}$$) ($$\overline{E}$$ is the closure of E) can be a proper subset of $$\overline{f(E)}$$ ? And is there something wrong with my attempt below?

The Attempt at a Solution

If E is compact, $$\overline{E}$$ = E, f(E) is compact, $$\overline{f(E)}$$ = f(E). Hence, f($$\overline{E}$$)= $$\overline{f(E)}$$
If E is not compact, $$\overline{E}$$ is closed and hence is compact, if E is bounded in R^{k}. f($$\overline{E}$$) is compact and hence $$\overline{f(\overline{E})}$$ = f($$\overline{E}$$).
since f(E) $$\subset{f(\overline{E})}$$ , $$\overline{f(E)}$$ $$\subset{\overline{f(\oveline{\overline{E}})}}$$ = f($$\overline{E}$$). It is also true that f($$\overline{E}$$) $$\subset{\overline{f(E)}}$$. Hence f($$\overline{E}$$) = $$\overline{f(E)}$$
In both cases, f($$\overline{E}$$) is not a proper subset of $$\overline{f(E)}$$

I've no idea of other kind of function that is continuous and with E otherwise defined.
Any hint would be greatly appreciated:)

HallsofIvy
Homework Helper

Homework Statement

If f is a continuous mapping of a metric space X into a metric space Y, Let E be any subset of X. How to show, by an example, that f($$\overline{E}$$) ($$\overline{E}$$ is the closure of E) can be a proper subset of $$\overline{f(E)}$$ ? And is there something wrong with my attempt below?

The Attempt at a Solution

If E is compact, $$\overline{E}$$ = E, f(E) is compact, $$\overline{f(E)}$$ = f(E). Hence, f($$\overline{E}$$)= $$\overline{f(E)}$$
If E is not compact, $$\overline{E}$$ is closed and hence is compact, if E is bounded in R^{k}. f($$\overline{E}$$) is compact and hence $$\overline{f(\overline{E})}$$ = f($$\overline{E}$$).
since f(E) $$\subset{f(\overline{E})}$$ , $$\overline{f(E)}$$ $$\subset{\overline{f(\oveline{\overline{E}})}}$$ = f($$\overline{E}$$). It is also true that f($$\overline{E}$$) $$\subset{\overline{f(E)}}$$. Hence f($$\overline{E}$$) = $$\overline{f(E)}$$
In both cases, f($$\overline{E}$$) is not a proper subset of $$\overline{f(E)}$$

I've no idea of other kind of function that is continuous and with E otherwise defined.
Any hint would be greatly appreciated:)
?? The problem was to show that f($$\overline{E}$$) can be a proper subset of $$\overline{f(E)}$$ and you have given two examples where that is NOT true. What's "wrong with that" is that you haven't answered the question!

Dick
Homework Helper
?? The problem was to show that f($$\overline{E}$$) can be a proper subset of $$\overline{f(E)}$$ and you have given two examples where that is NOT true. What's "wrong with that" is that you haven't answered the question!

Exactly. You want an example. There are perfectly fine examples from R->R. You don't need anything exotic. Can you find one? Hint, think about what happens if f has a horizontal asymptote.

to HallsofIvy,
I gave those two examples because I could not find a disired example :(

to Dick,
O I think I get it now...Let E=[1,$$\infty$$), f(x)=1/x and everything is done.....it is strange that I didn't get it...

Thanks a lot to all :)

Dick
O I think I get it now...Let E=[1,$$\infty$$), f(x)=1/x and everything is done.....it is strange that I didn't get it...