Pointwise convergence of holomorphic functions

[Radiator1]

Hello.

In my complex analysis book I've read a theorem which says that if a sequence $$\{ f_n \}$$ of holomorphic functions on a domain $$\Omega$$ converges pointwise to a function $$f$$, then $$f $$ is holomorphic on a dense, open subset of $$\Omega$$.

I know how to prove this theorem. I just find it hard to come up with an example of a sequence (as described above) for which the limit $f$ is not holomorphic on the entire $$\Omega$$.

 

[Euge]

Hello.

In my complex analysis book I've read a theorem which says that if a sequence $$\{ f_n \}$$ of holomorphic functions on a domain $$\Omega$$ converges pointwise to a function $$f$$, then $$f $$ is holomorphic on a dense, open subset of $$\Omega$$.

I know how to prove this theorem. I just find it hard to come up with an example of a sequence (as described above) for which the limit $f$ is not holomorphic on the entire $$\Omega$$.

Hi Radiator,

By application of Runge's theorem, you should be able to find a sequence of polynomials whose pointwise limit has a point of discontinuity.