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These questions were investigated by Osgood, Montel and Lavrentiev, Sur les fonctions d'une variable complexe representable par des series de polynomes, Paris 1936. (There is a Russian translation in his selected Works available free on Internet). If you prefer German, see Hartogs and Rosenthal, Uber Folgen analytischer Funktionen, Math Ann., 1928, 100, 212-263, and 1932, 104, 606-610.

In general, if a sequence of polynomials converges pointwise in a region $D$, then the limit function is analytic except for a closed nowhere dense set $E$ (Osgood). Montel proved that $E$ is a perfect set whose union with the complement of the disc is connected. Lavrentiev completely characterized the sets $E$ that can occur, and proved that every function of the first Baire class which is analytic outside $E$ is a pointwise limit of polynomials.

Thus every continuous function, analytic outside $E$ can be obtained as a limit of polynomials. Convergence outside $E$ is locally uniform.

By the way, similar problems for harmonic functions (characterization of their pointwise limits) is still not solved completely.

EDIT. For example, any simple curve in the unit disc, going from $0$ to $1$, satisfies the Lavrentiev condition. Taking this curve with positive area, we can construct a continuous function $f$ in the unit disc, which is analytic outside the curve but not analytic in the unit disc. This function will be a pointwise limit of polynomials.

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These questions were investigated by Osgood, Montel and Lavrentiev, Sur les fonctions d'une variable complexe representable par des series de polynomes, Paris 1936. (There is a Russian translation in his selected Works available free on Internet). If you prefer German, see Hartogs and Rosenthal, Uber Folgen analytischer Funktionen, Math Ann., 1928, 100, 212-263, and 1932, 104, 606-610.

In general, if a sequence of polynomials converges pointwise in a region $D$, then the limit function is analytic except for a closed nowhere dense set $E$ (Osgood). Montel proved that $E$ is a perfect set whose union with the complement of the disc is connected. Lavrentiev completely characterized the sets $E$ that can occur, and proved that every function of the first Baire class which is analytic outside $E$ is a pointwise limit of polynomials.

Thus every continuous function, analytic outside $E$ can be obtained as a limit of polynomials. Convergence outside $E$ is locally uniform.

By the way, similar problems for harmonic functions (characterization of their pointwise limits) is still not solved completely.

These questions were investigated by Osgood, Montel and Lavrentiev, Sur les fonctions d'une variable complexe representable par des series de polynomes, Paris 1936. (There is a Russian translation in his selected Works available free on Internet).

In general, if a sequence of polynomials converges pointwise in a region $D$, then the limit function is analytic except for a closed nowhere dense set $E$ (Osgood). Montel proved that $E$ is a perfect set whose union with the complement of the disc is connected. Lavrentiev completely characterized the sets $E$ that can occur, and proved that every function of the first Baire class function which is analytic outside $E$ is a pointwise limit of polynomials.

Thus every continuous function, analytic outside $E$ can be obtained as a limit of polynomials. Convergence outside $E$ is locally uniform.