Detect optional function argument (scalar)

Yes, you can do this:

function alfa(bravo, charlie) {  
  if (charlie) {
    return "charlie good"
  }
  if (charlie == 0 && charlie == "") {
    return "charlie not provided"
  }
  if (!charlie && charlie != 0) {
    return "charlie null"
  }
  if (!charlie && charlie != "") {
    return "charlie 0"
  }
}

Result:

charlie good
charlie 0
charlie null
charlie not provided

awk doesn't have a builtin way to check variable had been initialized. You must do it yourself:

function alfa(bravo, charlie) {
  if (charlie == 0 && !length(charlie))
    return "charlie bad"
  else {
    return "charlie good"
  }
}
BEGIN {
  print alfa(1, 1)
  print alfa(1, 0)
  print alfa(1, "")
  print alfa(1)
}

The general way:

var == 0 && !length(var)

An uninitialized variable and 0 causes var == 0 to be evaluated to true, !length(var) excludes the case when var = 0.