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Example of a complex function which is non constant and bounded in C?

Liouville's theorem states that if f(z) is holomorphic and bounded on all of the complex plane, then it is constant.

So to find an example of a function which is non-constant and bounded on all C, must it neccessarily be non-holomorphic so as not to contradict Liouville's theorem?

Would z/(|z|+1) be an example of such a function? (Bounded above by 1 and below by -1, and clearly not holomorphic)
(edited 11 years ago)
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Mark85
16
Yes. Here is another

Unparseable latex formula:

f(z) = \begin{cases} 0 &, x \mbox{ is rational} \\ 1 &, x \mbox{ is irrational}\end{cases}



where z=x+iy z= x+iy
(edited 11 years ago)

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