1

Let $A \subset \mathbb{R}^{n}$ be a bounded set and $f$ be a continuous function, $ f:\mathbb{R}^{n}\to\mathbb{R}^{m}$. How do I show that $\overline{f(A)} \subset f(\overline{A})$?

What I have tried so far: $A \subset \overline{A}$, so $f(A) \subset f(\overline{A})$ and $\overline{f(A)} \subset \overline{f(\overline{A})}$. Since, $A$ is bounded we have a closed ball $B$ such that $A \subset B$. I'm not sure where to go from here. Any suggestions?

PinkyWay
  • 4,565
Alpha not the centurion
  • 53
  • Try expanding the body of the question a bit with your thoughts. It will make it more clear to people how this differs from the proposed duplicate. – Alexander Gruber Apr 19 '20 at 02:03

1 Answers1

1

Your argument would be complete if you can show that $\overline{f(\overline{A})}=f(\overline{A})$, i.e. that $f(\overline{A})$ is closed. Certainly, $\overline{A}$ is closed, but continuous functions need not take closed sets to closed sets. However, they must take compact sets to compact sets and compact sets are closed. Is $\overline{A}$ compact? Hint: Heine-Borel.

Thorgott
  • 11,682