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 1 of 6: Set

x=1

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In plain words

This single substitution already links $f\circ f$ to the identity map, up to the constant $f(1)$ — a strong hint that $f$ should be close to an involution.

f(f(y)) = y f(1)

Detailed analysis

Substituting $x=1$ into (i) gives $f(f(y)) = y f(1)$ for every positive real $y$ .