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We define a continuous extension of $f(x)$ to the set $[a,b]$, by $g(x)=f(x), x\in (a,b)$ and $g(a)= \lim_{x \to a^{+}}f(x)$ and $g(b)=\lim_{x \to b^{-}}f(x)$. $g(x)$ being continuous on a compact set in $\mathbb{R}$, it is bounded. So, $f$ being a restriction of $g$, it must also be bounded.

Is this correct?

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  • $\begingroup$ I think you need to argue that the limits exist. If the limits exist, then this method works. $\endgroup$ – Clayton May 16 at 1:21
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    $\begingroup$ @Clayton they must. Because $f$ is uniformly continuous. (Proof is manageable). And after showing that the limits exist, then I move on to my procedure. Will that be alright? $\endgroup$ – Subhasis Biswas May 16 at 1:27
  • $\begingroup$ Correct! ${}{}{}$ $\endgroup$ – Clayton May 16 at 1:36
  • $\begingroup$ You can show that the limits exist but you can also show that $f$ is bounded without first showing that the limits exist. $\endgroup$ – DanielWainfleet May 16 at 2:47

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