Composing partial functions such that isDefinedAt works end-to-end. Requires that the functions are pure. Adapted from https://groups.google.com/forum/#!topic/scala-user/fnw_367xTTw

ComposePartial.scala

val f = Map("a" -> 1, "b" -> 2)                 //> f  : scala.collection.immutable.Map[String,Int] = Map(a -> 1, b -> 2)
val g = Map(1 -> 'c', 3 -> 'd')                 //> g  : scala.collection.immutable.Map[Int,Char] = Map(1 -> c, 3 -> d)

//def pandThen[A, B, C](pf1: PartialFunction[A, B], pf2: PartialFunction[B, C]): PartialFunction[A, C] = {
//  Function.unlift((a: A) => pf1.lift(a) flatMap pf2.lift)
//}                                              

implicit class ComposePartial[B, C](pf: PartialFunction[B, C]) {
    def andThenPartial[D](that: PartialFunction[C, D]): PartialFunction[B, D] =
      Function.unlift(pf.lift(_) flatMap that.lift)
    def composePartial[A](that: PartialFunction[A, B]): PartialFunction[A, C] =
      Function.unlift(that.lift(_) flatMap pf.lift)
}
  
val h = f andThenPartial g                     //> h2  : PartialFunction[String,Char] = <function1>

h.isDefinedAt("a")                             //> res6: Boolean = true
h.isDefinedAt("b")                             //> res7: Boolean = false
h.lift("a")                                    //> res8: Option[Char] = Some(c)
h.lift("b")                                    //> res9: Option[Char] = None