Pure functions are a fundamental concept in functional programming, characterized by their consistent output for the same input and the absence of side effects.

In detail, functional programming is a paradigm that views computation as the evaluation of mathematical functions, deliberately avoiding mutable data and changing states. The notion of a pure function is rooted in mathematics, where a function yields the same result whenever the input values remain unchanged, irrespective of how many times the function is invoked.

A pure function possesses two key properties:

1. Referential Transparency: A pure function always produces the same output for the same input. For instance, consider a function that takes two numbers, $a$ and $b$, as inputs and returns their sum, $a + b$. This function is pure because, whenever the same pair of numbers is provided, it consistently returns the same result.

2. No Side Effects: A pure function does not alter any state or data outside its own scope. It strictly operates with the provided input and does not modify any external variables or data structures. This characteristic enhances the predictability of pure functions, making them easier to test since their behavior does not depend on external states.

In contrast, an impure function may yield different outputs for the same input or exhibit side effects. For example, a function that retrieves the current system time is impure, as it produces varying results upon each call, even with identical input. Similarly, a function that modifies a global variable or writes data to a file is considered impure due to its side effects.

In functional programming, pure functions are favored for their ability to simplify reasoning, testing, and debugging of code. They also enable optimizations that are unattainable with impure functions. Nevertheless, it’s important to recognize that not every function in a functional programming language is pure. Impure functions may still be utilized when necessary, but their use is carefully managed and confined to preserve the advantages of functional programming.