lean_workbook_plus_35663

⚠️ 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 $f:(0,+\infty)\to(0,+\infty)$ so that $f(x)f(y)=f(xy)+f(\frac{x}{y}),\forall x,y\in (0,+\infty)$

Why this node was retired

The posted statement is

theorem lean_workbook_plus_35663 (f : ℝ → ℝ) (hf: Continuous f) (hf1: ∀ 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

Restoring the missing codomain condition alone does not fix the incorrect answer.

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 positivity of f on(0,∞), continuity on that domain, and the find-all-functions task. Remove the invented identity answer; constant2 is a positive solution.

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.