函數方程簡介
這份筆記介紹了一種在數學奧林匹亞競賽中經常出現的有趣題型 —— 函數方程。函數方程是一種未知數為一個或多個函數的方程。例如，考慮以下方程： \[ xf(x + f(y)) + xf(y) = 2yf(x) + x^2.\] 我們的任務是找出所有滿足該方程的函數 $ f : \mathbb{R} \to \mathbb{R} $。
在這份筆記中，我將提供一些指導，幫助你熟悉解決函數方程的常用技巧。需要注意的是，這份筆記的內容是用中文書寫的。
Introduction to Functional Equations
This note introduces a type of problem frequently encountered in Mathematical Olympiads —— functional equations. A functional equation is an equation where the unknowns are one or more functions. For example, consider the following: \[ xf(x + f(y)) + xf(y) = 2yf(x) + x^2. \] Here, the task is to find all functions $ f : \mathbb{R} \to \mathbb{R} $ that satisfy the equation.
In this note, I provide a guide to help you become familiar with the common techniques for solving functional equations. Please note that the content is written in Mandarin Chinese.