Let's take a high school example and try to explain the use of functions returning functions.

Our problem is we are given a cylinder of radius r and we would like to know the volume of it for different heights:

http://www.mathopenref.com/cylindervolume.html

Volume = area * height = pi * r^2 * h

The preceding formula gives the exact cubic units that will fill a cylinder.

Let's write a simple function to demonstrate the concept of a function returning a function. In addition, we will write a small piece of code to show the usage:

# 1.	Let us define a function which will explain our
#  concept of function returning a function.
def cylinder_vol(r):
    pi = 3.141
    def get_vol(h):
        return pi * r**2 * h
    return get_vol

# 2.	Let us define a radius and find get a volume function,
#  which can now find out the volume for the given radius and any height.
radius = 10
find_volume = cylinder_vol(radius)

# 3.	Let us try to find out the volume for different heights
height = 10
print "Volume of cylinder of radius %d and height %d = %.2f  cubic units" 
                %(radius,height,find_volume(height))        

height = 20
print "Volume of cylinder of radius %d and height %d = %.2f  cubic units" 
                %(radius,height,find_volume(height))        

In step 1, we defined a function called cylinder_vol(); it takes a single parameter, r, radius. In this function, we defined another function, get_vol(). The get_vol() function has access to r and pi, takes the height as an argument. For the given radius, r, which was the parameter to cylinder_vol(), different heights were passed as a parameter to get_vol().

In step 2, we defined a radius; in this case, as ten and invoke the cylinder_vol() function with it. It returns the get_vol() function, which we stored in a variable named find_volume.

In step 3, we invoked find_volume with different heights, 10 and 20. Note that we didn't give the radius.

The output produced is as follows:

Volume of cylinder of radius 10 and height 10 = 3141.00  cubic units
Volume of cylinder of radius 10 and height 20 = 6282.00  cubic units

Functools is a module for higher order functions:

https://docs.python.org/2/library/functools.html