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If $\phi: \mathbb{C}\to \mathbb{C}$ is uniformly continuous and $f_n \to f$, almost uniformly or in measure, then $\phi \circ f_n \to \phi \circ f$ almost uniformly, or in measure, respectively.

Find counterexamples when the continuity assumptions on $\phi$ are not satisfied.

I haven't been able to come up with counterexamples for these two cases. I can think of something when $\phi: R \to R$, but not in the complex case. Can anyone provide me with some examples? I would greatly appreciate any help.

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  • $\begingroup$ Is "almost uniform" here interpreted in the measure-theoretic sense (for every $\delta > 0$, uniform on the complement of a set of measure $< \delta$), or in the complex-analysis sense (uniform on compact sets)? $\endgroup$ – Robert Israel Mar 3 '16 at 21:45
  • $\begingroup$ in the measure theoretic sense. $\endgroup$ – nomadicmathematician Mar 3 '16 at 21:46
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Try $\phi(z) = z^2$, $f_n(z) = z + 1/n$, $f(z) = z$.

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