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What is an example of a continuous function on a closed and unbounded set with no maximum? Is $f(x)=x^3$ a correct example?

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$f(x)=x$ is perhaps a simpler example. – Gerry Myerson Nov 21 '12 at 5:38
up vote 4 down vote accepted

Yes, your example $f:\mathbb R\to\mathbb R,f(x)=x^3$ is correct, since $\mathbb R$ is closed in $\mathbb R$ and unbounded and $f\to +\infty$ as $x\to +\infty$.

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another good, concise answer – amWhy Nov 22 '12 at 0:04

Your example is correct. Simpler might be $f(x) = x$ and maybe more interesting $f(x) = x^{37} + x^{10} +1$, but not $f(x) = - x^2$ (why?)

Some other examples:

  • $f(x) = -e^{x}$: bounded above by $0$ but has no maximum.
  • $f(x) = -e^{-x^2}$: bounded above, assumes its minimum $-1$ at $0$ is bounded above by $0$ but has no maximum.
  • $f(x) = \arctan{x}$: bounded above and below by $\pm \frac{\pi}{2}$ but has neither maximum nor a minimum.

See the plots by Wolfram|Alpha.

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