Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If (1) $f$ is trueanalytic, is the convergence $\underline{locally}$ uniform on $\Delta$? (Note: I add the words "locally" due to obvious reason.)

If we do not assume that the limit function $f$ is continuous (of course $f$ is measurable) in advance, (3) can we say that $f$ is continuous? [I number this new question by (3)]

Note that evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).

Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If (1) is true, is the convergence $\underline{locally}$ uniform on $\Delta$? (Note: I add the words "locally" due to obvious reason.)

If we do not assume that the limit function $f$ is continuous (of course $f$ is measurable) in advance, (3) can we say that $f$ is continuous? [I number this new question by (3)]

Note that evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).

Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If (1) is true, is the convergence uniform on $\Delta$?

If we do not assume that the limit function $f$ is continuous (of course $f$ is measurable) in advance, (3) can we say that $f$ is continuous? [I number this new question by (3)]

Note that evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).

Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If (1) is true, is the convergence uniform on $\Delta$?

Note that evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).

Question: Let ${f_n}$ be a sequence of analytic functions on the unit disk $\Delta$ and suppose that $f_n$ converges to a continuous function $f$ on $\Delta$ pointwisely. (1) Can we say that $f$ is analytic on $\Delta$? (2) If (1) is true, is the convergence uniform on $\Delta$?

Note evrey measurbale functon on $\Delta$ can be the limit of a sequence of analytic functions in the Lebesgue sense (i.e. almost everywhere).