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Let $f:R\to R$ be continous function. Then, which one of the following sets can't be the image of $(0,1]$ under $f$.

$a) {0}$

$b) (0,1)$

$c) [0,1)$

$d) [0,1]$

the ans is $b$, but I don't know why, please explain it in detail.

Gaurav Agarwal
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Mrutyunjay Meher
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    You keyboard appears to be defective; it is omitting random letters in the words you type! (And somehow it seems to have made one of the letters in "please" into a "z"). – hmakholm left over Monica Jun 03 '17 at 15:39

1 Answers1

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Since $f$ is continuous on the whole $\Bbb R$, $f([0,1])=f((0,1])\cup\{f(0)\}$ must be compact. But $(0,1)\cup\{x\}$ is never compact for any $x\in\Bbb R$.

Of course, the diligent student should also come up with examples where $f((0,1])$ is each of $(a),(c),(d)$.