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

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

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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(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 arguments to 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

print(f(3))

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

print(plusOne(3))

4
```

```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

print(first([2,4,5]))
print(first([]))
```

output

```2

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

```head [1,2,3]
3

```

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)

[1, 2, 3, 4]
[1, 2, 3, 4, 5] [1, 2, 3, 4, 5]
```

```addElem :: a -> [a] -> [a]
addElem x xs = x : xs

=> [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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