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Does there exist a function $f:[0,1]\to [0,1]$ such that the graph of $f$ is dense in $[0,1]\times[0,1]$?

Not necessarily continuous.

  • 1
    The image is a subset of [0,1]. Then you want it to be dense in [0,1].2012-05-02
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    @alpha.Debi: Perhaps OP means the "graph" of $f$, so $$\{\big(x,f(x)\big):x\in[0,1]\}\subset [0,1]\times[0,1].$$2012-05-02
  • 0
    Yes, is in the graph, sorry. I have a mistake when I was typing.2012-05-02
  • 0
    Would [Hilbert curves](http://en.wikipedia.org/wiki/Hilbert_curve) be something of interest?2012-05-02
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    @Jean-Luc: No, those are (images of) functions $[0,1]\to[0,1]\times[0,1]$.2012-05-02

5 Answers 5