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Question: Show that there is no function from $\mathbb{R}$ to $\mathbb{R}$ such that for every real number a, $f(x)$ approaches infinity as $x$ approaches $a$.

We know that, limit of function $f$ as $x$ approaches to $a$ is infinity if for all $N>0$ there exist $\delta>0$ such that $f(x)>N$ whenever $|x-a|<\delta$.

then by above we get $f(x)>m$ for every $m\in \mathbb{R}$ and hence such real valued function $f$ does not exist.

is this proof is correct? If not Please explain me what details are missing and what mistakes I had done in proof. Thank you.

Akash Patalwanshi
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  • @David Mitra sir please elaborate and then what's the right way to proceed – Akash Patalwanshi Mar 03 '19 at 07:25
  • I don't see how you get your conclusion. Perhaps more detail is needed. (You need $0<|x-a|<\delta$ in the second sentence.) – David Mitra Mar 03 '19 at 07:45
  • Sir what more details are needed? and @RRL sir though this question was already answered. But as there are many ways to prove same result, so here too there may be many ways to solve above question. – Akash Patalwanshi Mar 03 '19 at 13:08

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