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Does there exist a function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the following?

  1. $f$ is unbounded above on every open interval.
  2. For every $x$, there exists an open interval $S$ containing $x$ such that for all $u\in S$, $f(x)\leq f(u)$.
  • 2
    With $\mathbb Q$ instead of $\mathbb R$ it would be easy: $\frac ab\mapsto b$.2012-11-25
  • 0
    Is there even a non-constant function satisfying 2.?2012-11-25
  • 1
    @Jens: Consider the function given by $f(x)=0$ if $x\leq 0$ and $f(x)=1$ if $x>0$.2012-11-25
  • 0
    *cough* True =)2012-11-26

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