Higher Order Function in Scala Explained. This example is taken from Martin Odersky's teaching at Coursera.

HigherOrderFunction.sc

package learn.scala

object Exercise {
	// Exercise on implementing Higher order functions in Scala.
	// All these examples are taken from Martin Odersky teachings on coursera.

  // EXERCISE 1:
  // Write a product function that calculate product of values of a function for the points
  // on a given interval

  def prod(f: Int => Int, a: Int, b: Int): Int = {
    if (a > b) 1 else {
      f(a) * prod(f, a + 1, b)
    }
  }                                               //> prod: (f: Int => Int, a: Int, b: Int)Int

  prod(x => x, 1, 5)                              //> res0: Int = 120

  // EXERCISE 2:
  // Write Factorial in terms of product

  def factorial(a: Int): Int = {
    prod(x => x, 1, a)
  }                                               //> factorial: (a: Int)Int

  factorial(10)                                   //> res1: Int = 3628800

  //==============================================================================================
  // EXERCISE 2:
  // finding a fixed point of function example
  // x, f(x), f(f(x)),f(f(f(x))),....
  // until value doesn't vary any more

  // first fixed point value f(x) = 1 + x/2
  //---------- my version
  def fixedPoint(f: Double => Double)(x: Double): Double = {
    def abs(y: Double): Double = if (y > 0) y else -y
    //println(x)
    if (abs(x - f(x)) <= 0.0000001) f(x) else (fixedPoint(f))(f(x))
  }                                               //> fixedPoint: (f: Double => Double)(x: Double)Double

  fixedPoint(x => 1 + x / 2)(1)                   //> res2: Double = 1.9999999403953552
  // find square root using this definition of fixed point.
  // definition of square root function interms of fixed point is as below:
  // sqrt(x) is a fixed point of the function y = x/y

  def sqrt(x: Double): Double = fixedPoint(y => (y + x / y) / 2)(1)
                                                  //> sqrt: (x: Double)Double
  sqrt(2)                                         //> res3: Double = 1.414213562373095
}