What is the point of an identity function if it's just the same as the original function?
Identity function serves as the base of all other functions. For example, if you are dealing with a function of functions, then the base case of this function may return an identity function.

Consider a simple function-generating function `g(n) = f^n; f^n(x) = x * n`. It is clear that `g(1) = f^1` is an identity function, since `f^1(x) = x * 1 = x` for all x. Here, identity function serves as the base function when we analyze `g` at different values of `n`.
What fields do Identity functions get used in apart from a maths classroom?
Identity functions can be found in many fields beyond mathematics, including computer science, physics, and engineering. Here are a few examples:

Computer Science: Identity functions are used in computer programming to transform data without changing its fundamental nature. For example, in Python, the built-in identity function id() returns the memory address of an object, and the lambda x: x function can be used to create a simple identity function.

Physics: In physics, identity functions can be used to represent the identity transformation of vectors or matrices. For example, the identity matrix in linear algebra is a square matrix in which all the elements of the main diagonal are 1 and all other elements are 0. This matrix can be used to transform vectors without changing their direction or magnitude.

Engineering: In engineering, identity functions can be used in control theory to represent the identity transformation of a system. For example, in a feedback control system, an identity function can be used to model a system in which the output is directly proportional to the input.

In general, identity functions are useful whenever there is a need to transform data without changing its fundamental properties.