Let f:R→R satisfy

f(a)≠0 for some a in R

f(xf(y))=yf(x) for all x,y in R
Prove that f(x)=f(x) for all x in R.
My solution is similar to JLo's:
1: proving f(0)=0
a) f(0*f(0))=0*f(0) which means f(0)=0
2: proving f(nonzero)=nonzero
a) assume f(b)=0 for some nonzero b
b) f(a*f(b))=b*f(a) which means f(0)/b=f(a)
c) f(0)/b=0, so f(a)=0
d) this contradicts condition 1, so the assumption must be false
3: proving f(1)=1
a) let f(1)=c
b) f(1*f(1))=1*f(1) which means f(c)=c
c) f(c*f(1))=1*f(c) which means f(c*c)=c
d) f(c*f(c))=c*f(c) which means f(c*c)=c*c
e) since c=c*c and f(nonzero)=nonzero, c=1, thus f(1)=1
4: proving f(1)=1
a) let f(1)=d
b) f(1*f(1))=1*f(1) which means f(d)=1
c) f(1*f(d))=d*f(1) which means f(1)=d*d and d=1 or 1
d) f(d*f(d))=d*f(d) which means f(d)=d
e) if d=1, then f(1)=1, but f(1)=d=1
f) this leads to a contradiction so d=1, and f(1)=1
5: proving f(x)=f(x)
a) f(x*f(1))=1*f(x) which means f(x)=f(x) since f(1)=1

Posted by Gamer
on 20061229 00:14:29 