“Unit tests are small tests, each one exercising a little piece of functionality. The units tested are usually individual methods, but sometimes clusters of methods or even whole objects.”
Extreme Programming Explained: Embrace Change, 2nd Edition (2004) Kent Beck
You're agreeing with me there. "each one exercising a little piece of functionality". In Beck's approach the tests are added at each step for a simple functionality addition, which then gets implemented in the project to pass the test. These days unit tests are more commonly used as "every method is a unit to be tested completely", often after the implementation is already there.
“Unit tests are small tests, each one exercising a little piece of functionality. The units tested are usually individual methods, but sometimes clusters of methods or even whole objects.”
Extreme Programming Explained: Embrace Change, 2nd Edition (2004) Kent Beck