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- Homework Statement
- .
- Relevant Equations
- complex analysis
##f## is continuou on ##\mathbb{C}##, so for al ##\epsilon>0##, there is a ##\delta>0## such that $$|\tilde{z}-z|\leq \delta \Rightarrow |f(\tilde{z})-f(z)|\leq \epsilon$$ for all ##\tilde{z}## and ##z## in ##\mathbb{C}##.
Complex conjugation is a norm preserving operation on ##\mathbb{C}##, so
$$|f(\tilde{z})-f(z)|=|\bar{f}(\tilde{z})-\bar{f}(z)|\leq \epsilon$$
The same ##\delta## satisfies the definition of continuity for ##\bar{f}##, so ##\bar{f}## is continuous on ##\mathbb{C}##.
The product of two continuous functions is continuous, so ##g=f\cdot \bar{f}## is continuous.
The conclusion feels shaky or like I'm not doing the real work of the problem somehow.
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