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What is the point of an identity function if it's just the same as the original function?
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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`.
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What fields do Identity functions get used in apart from a maths classroom?
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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.