## What exactly is meant by "partial function" in functional programming?

According to my understanding, partial functions are functions that we get by passing fewer parameters to a function than expected. For example, if this were directly valid in Python:

>>> def add(x,y): ... return x+y ... >>> new_function = add(1) >>> new_function(2) 3

In the snippet above, `new_function`

is a partial function. However, according to the Haskell Wiki, the definition of partial function is

A partial function is a function that is not defined for all possible arguments of the specified type.

so, my question is: what exactly is meant by "partial function"?

You are here confusing two concepts. A *partially applied function* [haskell-wiki] with a

*partial function*[haskell-wiki].

A partially *applied* function is:

Partial application in Haskell involves

passing less than the full number of argumentsto a function that takes multiple arguments.

whereas a partial function indeed is a non-total function:

A partial function is a function that is not defined for all possible arguments of the specified type.

**"What exactly" vs. "Exactly what",** describes and adjective (and consequently it NEVER describes a noun, such as the subject or object of a sentence.) A number or quantity having a value that is intermediate between other numbers or quantities, especially an arithmetic mean or average. See more at arithmetic mean.

A partial function (both in the context of functional programming and mathematics) is exactly what the wiki says: a function not defined for all of its possible arguments. In the context of programming, we usually interpret "not defined" as one of several things, including undefined behaviour, exceptions or non-termination.

An example of a partial function would be integer division, which is not defined if the divisor is 0 (in Haskell it will throw an error).

in above snippet new_function is partial function.

That code would simply cause an error in Python, but if it worked as you intended, it would be a total (meaning not partial) function.

As commentors already pointed out, you're most likely thinking of the fact that it'd be a *partially applied* function.

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The answers explain all, I will just add one example in each language:

def add(x,y): return x+y f = add(1) print(f(3)) f = add(1) TypeError: add() missing 1 required positional argument: 'y'

this is **neither a partial function nor a curried function**, this is only a function that you **didn't gave all its arguments**.

A curried function in python should be like this:

partialAdd= lambda x: lambda y: x + y plusOne = partialAdd(1) print(plusOne(3)) 4

and in haskell:

plus :: Int -> Int -> Int plus x y = x + y plusOne = plus 1 plusOne 4 5

A partial function in python:

def first(ls): return ls[0] print(first([2,4,5])) print(first([]))

output

2 print(first([])) File "main.py", line 2, in first return ls[0] IndexError: list index out of range

And in Haskell, as your link showed up:

head [1,2,3] 3 head [] *** Exception: Prelude.head: empty list

So what is a total function?

Well, basically the opposite: this is a function that will work for any input of that type. Here is an example in python:

def addElem(xs, x): xs.append(x) return xs

and this works even for infinite lists, if you use a little trick:

def infiniList(): count = 0 ls = [] while True: yield ls count += 1 ls.append(count) ls = infiniList() for i in range(5): rs = next(ls) print(rs, addElem(rs,6)) [1, 2, 3, 4] [1, 2, 3, 4, 5] [1, 2, 3, 4, 5]

And the equivalent in Haskell:

addElem :: a -> [a] -> [a] addElem x xs = x : xs addElem 3 (take 10 [1..]) => [3,1,2,3,4,5,6,7,8,9,10]

Here the functions doesn't hang forever. The concept is the same: for every list the function will work.

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