Real and Complex Analysis (Rudin)

ELEMENTARY PROPERTIES OF HOLOMORPHIC FUNCTIONS 115
PROOF The only points (z, w) E n x n at which the continuity of g is possibly
in doubt have z = w.
Fix a E n. Fix E > O. There exists r > 0 such that D(a; r) c: nand
I I'm - f'(a) I < E for all , E D(a; r). If z and ware in D(a; r) and if
then W) E D(a; r) for 0 :s; t :s; 1, and
g(z, w) - g(a, a) = r
W) = (1 - t)z + tw,
[f'(W)) - f'(a)] dt.
The absolute value of the integrand is 0 so that D(a, r) c: V. By (1)
there exists c > 0 such that
I qJ(a + rei~ - qJ(a) I > 2c ( -11: :s; () :s; 11:).
If A. E D(qJ(a); c), then I A. - qJ(a) I < c, henCe (3) implies
min IA. - qJ(a + rei~1 > c.
8
By the corollary. to Theorem 10.24, A. - qJ must therefore have a zero in
D(a; r). Thus A. = qJ(z) for some Z E D(a; r) c: V.
This proves that D(qJ(a); c) c: qJ(V). Since a was an arbitrary point of V,
qJ( V) is open.
To prove (c), fix WI E W. Then qJ(ZI) = WI for a unique ZI E V. If WE W
and ",(w) = Z E V, we have
(1)
(2)
(3)
(4)
",(w) - "'(WI) Z - ZI
W - WI qJ(Z) - qJ(ZI)"
(5)