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## 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([tex]\overline{E}[/tex]) ([tex]\overline{E}[/tex] is the closure of E) can be a proper subset of [tex]\overline{f(E)}[/tex] ? And is there something wrong with my attempt below?

## Homework Equations

## The Attempt at a Solution

If E is compact, [tex]\overline{E}[/tex] = E, f(E) is compact, [tex]\overline{f(E)}[/tex] = f(E). Hence, f([tex]\overline{E}[/tex])= [tex]\overline{f(E)}[/tex]

If E is not compact, [tex]\overline{E}[/tex] is closed and hence is compact, if E is bounded in R^{k}. f([tex]\overline{E}[/tex]) is compact and hence [tex]\overline{f(\overline{E})}[/tex] = f([tex]\overline{E}[/tex]).

since f(E) [tex]\subset{f(\overline{E})}[/tex] , [tex]\overline{f(E)}[/tex] [tex]\subset{\overline{f(\oveline{\overline{E}})}} [/tex] = f([tex]\overline{E}[/tex]). It is also true that f([tex]\overline{E}[/tex]) [tex]\subset{\overline{f(E)}}[/tex]. Hence f([tex]\overline{E}[/tex]) = [tex]\overline{f(E)}[/tex]

In both cases, f([tex]\overline{E}[/tex]) is not a proper subset of [tex]\overline{f(E)}[/tex]

I've no idea of other kind of function that is continuous and with E otherwise defined.

Any hint would be greatly appreciated:)