Problem 1

Find all functions

f

defined on the set of positive real numbers which take positive real values and satisfy the two conditions: (i)

f(xf(y)) = yf(x)

for all positive real numbers

x, y

; (ii)

f(x) \to 0

x \to \infty

Step 4 of 6: Fixed points multiply, and invert

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f(a)=a,\ f(b)=b \ \Rightarrow\ f(ab)=ab,\quad f\!\left(\tfrac1a\right)=\tfrac1a

Detailed analysis

Suppose $f(a)=a$ and $f(b)=b$ . Setting $x=a,y=b$ in (i): $f(ab)=f(af(b))=bf(a)=ab$ , so $ab$ is also a fixed point. Setting $x=1/a, y=a$ : $1=f(1)=f(\tfrac1a f(a))=a f(\tfrac1a)$ , so $f(1/a)=1/a$ : the reciprocal of a fixed point is a fixed point.