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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor series in $\Delta$ which has $R=1$ as its radius of convergence. We further assume that $f(z)$ is differentiable (complex-)differentiable at every point of $S^1$. (where the differentiability is global at a point instead of only along certain directions in $\overline\Delta$.) Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.

Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?

Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ uniform or absolute on $\overline\Delta$? (Note."uniform" does not imply "absolute".)

Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this): http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series

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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor series in $\Delta$ which has $R=1$ as its radius of convergence. We further assume that $f(z)$ is differentiable at every point of $S^1$. (where the differentiability is global at some a point instead of only along certain directions in $\overline\Delta$.) Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.

Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?

Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ uniform or absolute on $\overline\Delta$? (Note."uniform" does not imply "absolute".)

Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this): http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series

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For simplicity, we assume that $f(z)$ is analytic in the unit disk $\Delta: |z|<1$ and continous in the closed disk $\overline \Delta$. Let $f(z)=\sum a_nz^n$ be its Taylor seris series in $\Delta$ which has $R=1$ as its radius of convergence. We further assume that $f(z)$ is differentiable at every point of $S^1$. (where the differentiability is global at some point instead of only along certain directions in $\overline\Delta$.) Such a function $f(z)$ exists. Please see a simple example.
Let $f(z)=(1-z)^2\log (1-z)$.

Question 1. Is the Taylor series $\sum a_nz^n$ convergent at every point of $S^1$?

Question 2. If the answer to Question 1 is yes, is the convergence of $\sum a_nz^n$ is uniform or absolute on $\overline\Delta$? (Note."uniform" doesnot does not imply "absolute".)

Notice. There is a function $f$ analytic in $\Delta$ and continous in $\overline\Delta$ such that its Taylor series $\sum a_nz^n$ is divergent at some pont in $S^1$. Please see (who can teach me how to cite a simple super-link to this): http://math.stackexchange.com/questions/286119/continuity-of-analytic-function-implies-convergence-of-power-series

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