Lua - Higher-order Derivative function

Question

How does returning functions work in Lua? I'm trying to wrap my head around this derivative function but I'm having trouble understanding it.

function derivative(f, delta)
  delta = delta or 1e-4
  return function(x)
      return (f(x + delta) - f(x))/delta
    end
end

Any help is appreciated, thanks.

Answer 1

First see here.

Shortly, functions are first-class citizens and you an store them in variable and return from functions.

Second. In your example there is a closure to be created. It will have upvalues f and delta, wich can be used in inner function.

So when you call your derivative function, new closure will be created with copy of f and delta. And you can call it later like any other function

local d = derivative(f, 1e-6)
d(10)

EDIT: "Answer on but I'm having trouble understanding how the x argument is treated in the anonymous function in my example"

Each function have a signature, number of formal attributes, it will get. In Lua you can call function with any number of arguments. Let's consider an example.

local a = function(x1, x2) print(x1, x2) end
a(1) // 1, nil
a(1, 2) // 1, 2
a(1, 2, 3) // 1, 2

When you call function in variable a, each given argument value, one by one will be matched with function argumentList. In 3-rd example 1 will be assigned to x1, 2 to x2, 3 will be thrown away. In term's of vararg function smth like this will be performed.

function a(...)
  local x1 = (...)[1]
  local x2 = (...)[2]
  // Body
end

In your example x is treated as inner function argument, will be visible inside it, and initialized when you call your inner function instance.

f and delta will be unique for each function instance, as I mentioned above.

Hope my clumsy explanations will hit their goal and help you a little.