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“Densely (un)Defined Function” graph Densely (un)Defined Function
A graph of the function xx; although the graph does not immediately appear to be a function, closer examination will reveal its true nature. Most graphing programs do not define the exponential function for negative bases.

For negative x, xx is

  • undefined if x is irrational,

  • undefined if x = odd / even,

  • a well-defined positive real number if x = even / odd, and

  • a well-defined negative real number if x = odd / odd.