Show that the boundary of any set D C is closed Suppose that

Show that the boundary of any set D C is closed. Suppose that f and g are analytic functions on D, and open connected subset of C. Prove that f/s is analytic at any point z_0 such that g(z_0) notequalto 0. Calculate 4/dz (f/g).

Solution

Here we will show that cl(D) contains all its boundary points, i.e. bd(cl(D))cl(D). Now from basic set theory we have the concept that it implies ybd(cl(D))ycl(D).
Let ybd(cl(D)). Now we show that ycl(D). By contradiction, ycl(D). Thus, y(Dbd(D)) and thus yD and ybd(D). It results yint(Dc).
Since ybd(cl(D)), for every open ball B(y) it should be that B(y)cl(D). It follows that either B(y)D or B(y)bd(D). Since yint(Dc), we find an where this is not true, a contradiction. Hence, we conclude that ycl(D).

Thus it proves the given statement.