The function of double underscore back and front in a class function name?

[snippsat]

To give a eaiser example then is easier to see how special method work.
They give extra behavior to a class,in this case can now use in operator and len().

class MyList:
    def __init__(self, data):
        self.data = data

    def __contains__(self, item):
        return item in self.data

    def __len__(self):
        return len(self.data)

Use Class:

>>> my_list = MyList([1, 2, 3, 4, 5])
>>> my_list.data
[1, 2, 3, 4, 5]
>>> # Now will "in" work,because of __contains__
>>> print(3 in my_list)
True
>>> print(9 in my_list)
False
>>> # len() works because of __len__
>>> len(my_list)
5

In summary, by defining the __contains__ method,
the MandelbrotSet class allows you to use the in operator to check if a complex number is part of the Mandelbrot set.
This makes it easy to check membership in the set with syntax like if c in mandelbrot_set.