Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 4, Problem 1

Problem

If $f$ is a continuous mapping of a metric space $X$ into a metric space $Y$, prove that

$$f(\overline{E}) \subset \overline{f(E)}$$

for every set $E \subset X$. ($\overline{E}$ denotes the closure of $E$.) Show, by an example, that $f(\overline{E})$ can be a proper subset of $\overline{f(E)}$.

Answer

For $y \in f(\overline{E})$, there exists $x \in \overline{E}$ such that $y=f(x)$. For $\epsilon > 0$, there exists $\delta > 0$ such that $d_X(x,x')<\delta$  $\Rightarrow$  $d_Y(f(x),f(x'))<\epsilon$. Since $x \in \overline{E}$, there exists $z \in E$ such that $d_X(x,z) < \delta$, which implies $f(z) \in N_\epsilon(f(x))$. This means $y=f(x) \in \overline{f(E)}$.

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