Existence of a piecewise power function

Proved

WorkbookRestored.plus_19163

by wamlart · Sep 12, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

lean-workbookrestored-importssource-checked

For $a>0$ and $t\ge0$ , there is a real function $g$ such that $g(x)=x^t$ for $x>0$ , $g(0)=0$ , and $g(x)=-a(-x)^t$ for $x<0$ .

This restores the missing imports or namespaces of the original node, retaining its formal proposition. Source: Lean-Workbook row lean_workbook_plus_19163 (Apache-2.0).

Preamble

import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Analysis.SpecialFunctions.Log.Basic
open Real

Formal statement

theorem WorkbookRestored.plus_19163 (a t : ℝ) (ha : a > 0) (ht : t ≥ 0) : ∃ g : ℝ → ℝ, ∀ x > 0, g x = x ^ t ∧ g 0 = 0 ∧ ∀ x < 0, g x = -a * (-x) ^ t   :=  by sorry

Source

https://huggingface.co/datasets/internlm/Lean-Workbook/blob/main/lean_workbook.json, row lean_workbook_plus_19163; original Prove2Me node d4eab463-c6a2-49a8-bd9a-cd8ae76e9f14; Apache-2.0

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