Using partial functions in Scala - how does it work?

A partial function is a function that is valid for only a subset of values of those types you might pass in to it. For example:

val root: PartialFunction[Double,Double] = {
  case d if (d >= 0) => math.sqrt(d)
}

scala> root.isDefinedAt(-1)
res0: Boolean = false

scala> root(3)
res1: Double = 1.7320508075688772

This is useful when you have something that knows how to check whether a function is defined or not. Collect, for instance:

scala> List(0.5, -0.2, 4).collect(root)   // List of _only roots which are defined_
res2: List[Double] = List(0.7071067811865476, 2.0)

This is not going to help you place two arguments where you really want one.

In contrast, a partially applied function is a function where some of its arguments have already been filled in.

def add(i: Int, j: Int) = i + j
val add5 = add(_: Int,5)

Now you only need one argument--the thing to add 5 to--instead of two:

scala> add5(2)
res3: Int = 7

You can see from this example how to use it.

But if you need to specify those two arguments, this still won't do it--say you want to use map, for instance, and you need to give it a function of one argument, but you want it to add two different things. Well, then you can

val addTupled = (add _).tupled

which will partially apply the function (really, just create a function out of the method, since nothing has been filled in) and then combine the separate arguments into a tuple. Now you can use this in places that require a single argument (assuming that the type is correct):

scala> List((1,2), (4,5), (3,8)).map(addTupled)
res4: List[Int] = List(3, 9, 11)

In contrast, currying is different yet again; it turns functions of the form (A,B) => C into A => B => C. That is, given a function of multiple arguments, it will produce a chain of functions that each take one argument and return a chain one shorter (you can think of it as partially applying one argument at a time).

val addCurried = (add _).curried

scala> List(1,4,3).map(addCurried)
res5: List[Int => Int] = List(<function1>, <function1>, <function1>)

scala> res5.head(2)   // is the first function, should add 1
res6: Int = 3

scala> res5.tail.head(5)   // Second function should add 4
res7: Int = 9

scala> res5.last(8)  // Third function should add 3
res8: Int = 11

Rex Kerr's explanation is very good -- and no surprise there either. The question is clearly mixing up partial functions with partially applied functions. For whatever it is worth, I made the same confusion myself when I learned Scala.

However, since the question does draw attention to partial functions, I'd like to speak of them a bit.

Many people say that partial functions are functions that are not defined for all input, and that's true of mathematics, but not of Scala. In Scala, a function may not be defined for all input either. In fact, since partial function inherit from function, then function includes all partial functions as well, making that inevitable.

Others mention the method isDefinedAt, which is, indeed, a difference, but one mostly about implementation. That's so true that Scala 2.10 will probably be released with a "fast partial function", which does not rely on isDefinedAt.

And a few people even imply that the apply method for partial functions do something different than the apply method for functions, like only executing for the input that is defined -- which could not be farther from the truth. The apply method is exactly the same.

What partial functions really come down to is another method: orElse. That sums up all use cases for partial functions much better than isDefinedAt, because partial functions are really about doing one of the following things:

• Chaining partial functions (which is what orElse does), so that an input will be tried on each partial function until one of them matches.
• Doing something different if a partial function doesn't match, instead of throwing an exception, which is what would happen if you chain that different thing using orElse.

I'm not saying everything can be easily implemented in terms of orElse, mind you. I'm just saying that partial functions are about doing something else when an input isn't defined for it.