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Finding a continuous and bounded function that attains its infimum
Let $u$ be a nonnegative measurable function in the whole $\mathbb{R}^n$ that only assumes the values zero and infinity. Let
$$M=\inf_{x\in\mathbb{R}^n}u(x).$$
Find a continuous and bounded function $f$ with the property that
$$\lim_{x\to\infty}f(x)=+\infty$$
and
$$\lim_{x\to-\infty}f(x)=-\infty.$$
(this problem from the book “Analysis” by Kreyszig)
The first idea i got is to use the Cantor-Schröder-Bernstein theorem. So to prove the existence i have to find a continuous and bounded function $g$ and a function $v$ such that:
$$\lim_{x\to\infty}g(x)=+\infty$$
and
$$\lim_{x\to-\infty}g(x)=-\infty.$$
Then $g+v$ satisfies the conditions of the theorem. But it does not really worked well as it is a variation of the Schauder theorem. So what if the functions I try to find are not bounded?
Thank you in advance,
Mateusz
A:
The required function $f$ can be obtained by a selection in the following way. By the definition of $M$, for every $\epsilon>0$ there exists $R>0$ such that $$M-\epsilon0$ be a real number such that $0
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