【函数概念并不难，理解“函”字是关键——函数概念如何理解】

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清. 李善兰 翻译 Function 为函数。函，信也。只能有一个收信人，所以 只有一个 f(x) 值。

…

The unique 1 single output of a function becomes very important for subsequent development in Math & IT:

functions are composable, associative, identify function，etc (distributive,… ) => it can be treated like vector => structure of a Vector Space “Vect”

Extended to..

“Vect” is a bigger structure “Category” in which “function of functions” is a

“Functor” （函子）F：F(f)

Example : F(f) = fmap (in Haskell)

fmap (+1) {2,7,6,3}

=> {3,8,7,4}

here F = fmap, f = +1

The Math branch in the study of functions is called “functional” 泛函。

IT : Functional Programming in Lisp, Haskell, Scala, ensure safety of guaranteed output by math function property. Any unexpected exception (side effects: IO, errors) is handled by a special function called “Monad” (endo-Functor).

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