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I'm reading one of the articles written by Sarason. He define the space $QC$ as the $C^*$algebra generated by $H^{\infty} + C$, that is, $QC=(H^{\infty} + C) \cap (\bar{H}^{\infty} + C)$. I'm using the fact that $QC\neq C$ but i need an explicit example of this. All that I have is this: if $f$ is a conformal maping between the unit disk and a "suitable" domain, then, the boundary function is in $QC$. Another possibility is to find a continuous real value function $u$ such that its conjugate $v$ (the function $v$ such that $u+iv$ is analytic) is discontinuous at one point.

I really appreciate your help. Thanks in advance.

J. M. ain't a mathematician
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Breitner Ocampo Gomez
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Another possibility is to find a continuous real value function $u$ such that its conjugate $v$ is discontinuous at one point.

That would not be enough, since discontinuity does not rule out membership in $H^\infty$. What you need is an example of a continuous function for which the conjugate is unbounded at a particular boundary point. I gave such an example here: it's based on a conformal map onto a domain with thin infinite tail, like $0<y<1/(1+x^2)$.

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