Abel's functional equation

Abel's functional equation involves finding a function \(f\) such that adding a fixed value \(c\) to the input of \(f\) corresponds to applying \(f\) to the original input and then adding a constant \(k\). Mathematically, it’s expressed as \(f(x + c) = f(x) + k\). This equation describes functions with a uniform rate of change over shifts, capturing patterns like linear functions. It’s useful in various fields, including dynamics and differential equations, for analyzing how functions behave under shifts or translations. Essentially, it links the behavior of a function when its input is increased by a constant.