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Assume that the metric space $(X, d)$ is not compact. Show that there exists $f: X\to \Bbb R$ which is continuous but not bounded.

The only function in my mind is $f: X\to \Bbb R$ defined by $f(x)=\|x\|$. I know this function is continuous. I don't know if it is bounded or not. I need a hint how to choose this function.

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Sunit das
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    Actually first i thought they want a example. I know few examples like that. But i have a confusion that they are asking for something general. – Sunit das Jun 18 '20 at 14:34
  • Looks like your answer is here: https://math.stackexchange.com/questions/1932306/unbounded-continuous-function-on-non-compact-metric-space – halrankard Jun 18 '20 at 14:40
  • Actually more answers are here: https://math.stackexchange.com/questions/1244557/existence-of-a-continuous-function-which-does-not-achieve-a-maximum – halrankard Jun 18 '20 at 14:41
  • I'm not sure what the function $f(x) = | x|$ is, as we are in a metric space, and that notation is typically used for the norm on a normed vector space. However, I think you are on the right track. We certainly have one function: $d:X \times X \to \mathbb{R}$, but this is a function of two variables. Can you use it to produce a function of one variable, and then maybe show that that function is not bounded? – Brian Shin Jun 18 '20 at 14:42

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