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My question is as follows: I have to choose true or false. This is a question from a Graduate school admission test.

Let $X$ be a connected metric space. Consider $F$ to be a subring of $C(X, \mathbb{R}) $ which is a field. Then prove that every element of $C(X, \mathbb{R}) $ that belongs to $F$ is constant.

I don't know the answer.

  • What have you tried? What does $F$ being a field tell you? – Florian R Dec 23 '21 at 10:33
  • Hint: 1) suppose that $f$ is not zero in $F$. Show that $f$ does not vanish. 2) Shows that constant maps are in $F$. 3) Show that if $f$ is in $F$, then by 1) and 2), $f$ is constant. – Didier Dec 24 '21 at 15:24

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