15/05/01 05:03:25.52 kLen/OPb.net
>>258 つづき
THEOREM 4. If φ is a wild automorphism of C then φ is a discontinuous mapping of the complex plane onto itself;
in fact，φ leaves a dense subset of the real line pointwise fixed but maps the real line onto a dense subset of the plane.
Proof. By Theorem 2，φ leaves Q (a dense subset of the real line!) pointwise fixed.
By Theorem 3 we can choose b∈R such that φ(b) not∈R.
Every neighborhood of b contains a rational number (which is left fixed by φ) and the number b (which is moved by φ); hence φ is discontinuous.
For every pair of rational numbers， q and r，φ(rb+q) =φ(r)φ(b)+φ(q)=rφ(b) +q.
Thus for a fixed r，{φ (rb+q) | q∈ Q} is a set of images of real numbers which is a dense subset of the horizontal line through rφ(b).
As r varies this horizontalline moves up and down; moreover the various rφ(b) form a dense subset of the(nonhorizontal)line through 0 and φ(b).
Thus the set {φ(rb+q) | r,q∈Q} is a dense subset of the plane.
This set is contained in φ(R); hence φ(R) is also a dense subset of C.
（引用おわり）