lean_workbook_plus_56705

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Find all continuous functions $f: (0, \infty) \to (0,\infty)$, for which $f(x)f(y) = f(xy) + f(x/y)$ for any positive $x$ and $y$.

Why this node was retired

The posted statement is

theorem lean_workbook_plus_56705 (f : ℝ → ℝ) (hf: Continuous f) (hx: ∀ x y : ℝ, (x > 0 ∧ y > 0) → f x * f y = f (x*y) + f (x/y)) : ∀ x : ℝ, (x > 0) → f x = x   :=  by sorry

A complete replacement classification is not established by the counterexample alone.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

Restore the classification task on the positive domain with positive codomain and continuity there; remove f(x)=x as the alleged unique answer. The constant2 must be included.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.