Re: Examples of two-to-one functions


"Vassakip Shatrumann" <shatrumann@xxxxxxxx> wrote in message
news:2821751.1192431185327.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
The function f:R->R, f(x)=x^2 is continuous and exactly two-to-one on
R\{0} (that is, the equation f(x)=c for all c in R\{0} has exactly two
different solutions).

Further, the function g:C->C, g(z)=z^2 (or g:R^2->R^2, g(x, y)=(x^2-y^2,
2xy)) is continuous and exactly two-to-one on C\{0} (resp. on R^2\{0}).

Let now n be any positive integer, n>2. Can we think up a function
F:R^n->R^n which is continuous and exactly two-to-one on R^n\{0}?

Great thanks!

On R,

abs(x), cosh(x), etc...

Take any function f(x) which is strictly monotonic on [0, oo)

then

g(x) = f(-x) if x < 0 and f(x) if x > 0

And more general if given two monotonic functions, you can piece them
together such as

h(x) = g(x) if x < 0 and f(x) if x > 0

where g(x) is monotonic on [-oo, 0] and sign(g'(x)) = sign(f'(x))


But that leads to the general case:

Let f(x) be a function s.t. f'(x) > 0 (< 0) for x < 0 and f'(x) < 0(> 0) for
x > 0, then f(x) is two-to-one except at 0.

This can be generalized to R^n quite easily.

It should be quite obvious why it must hold. The function cannot have any
critical points except at 0(which is exactly why you excluded zero) and the
derivatives on each side of 0 must be opposite or else its strictly
monotonic on (-oo,oo)







.

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