If $\varphi$ is a continuous function defined on the unit circle $T =\{z \in \mathbb{C} : |z| = 1 \}$, then $f (t) = \varphi(e^{it})$ is a continuous periodic function with period $2\pi$. Conversely, if $f$ is a continuous function on the real line of period $2\pi$, there exists a unique continuous function $\varphi$ defined on the unit circle $T$ such that $f (t) = \varphi(e^{it})$. Is my guess correct? Can anyone lead me a proof for this?
Continuous function on the circle and real line
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real-analysis
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0How continuous is "continuous"? – 2011-05-15
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1To detail a little bit J.M.'s question, because I wonder too : when you say $\varphi$ is a continuous function defined on the circle, you don't specify its image... what are the domains and codomains of your functions? After that it'll be easier to tell. – 2011-05-15
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0Well in Artin's Algebra there is a Lemma: The continuous homomorphisms $\varphi: \mathbb{R}^{+} \to U_{1}$ are of the form $\varphi(x)=e^{icx}$ for some $c \in \mathbb{R}$. – 2011-05-15
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0@user10805: you are right. Surely there is a unique function $\phi$ such that $f(t)=\phi(e^{it})$, so you want to know why $\phi$ is continuous. The map $\mathbb{R}\to T$, $t\mapsto e^{it}$, has locally a homeomorphism, so $\phi$ is continuous iff $f$ is. – 2011-05-15