Here is the snippet:

test = pd.DataFrame({'days': [0,31,45]})
test['range'] = pd.cut(test.days, [0,30,60])

Output:

    days    range
0   0       NaN
1   31      (30, 60]
2   45      (30, 60]

I am surprised that 0 is not in (0, 30], what should I do to categorize 0 as (0, 30]?

test['range'] = pd.cut(test.days, [0,30,60], include_lowest=True)
print (test)
   days           range
0     0  (-0.001, 30.0]
1    31    (30.0, 60.0]
2    45    (30.0, 60.0]

See difference:

test = pd.DataFrame({'days': [0,20,30,31,45,60]})

test['range1'] = pd.cut(test.days, [0,30,60], include_lowest=True)
#30 value is in [30, 60) group
test['range2'] = pd.cut(test.days, [0,30,60], right=False)
#30 value is in (0, 30] group
test['range3'] = pd.cut(test.days, [0,30,60])
print (test)
   days          range1    range2    range3
0     0  (-0.001, 30.0]   [0, 30)       NaN
1    20  (-0.001, 30.0]   [0, 30)   (0, 30]
2    30  (-0.001, 30.0]  [30, 60)   (0, 30]
3    31    (30.0, 60.0]  [30, 60)  (30, 60]
4    45    (30.0, 60.0]  [30, 60)  (30, 60]
5    60    (30.0, 60.0]       NaN  (30, 60]

Or use numpy.searchsorted, but values of days has to be sorted:

arr = np.array([0,30,60])
test['range1'] = arr.searchsorted(test.days)
test['range2'] = arr.searchsorted(test.days, side="right") - 1
print (test)
   days  range1  range2
0     0       0       0
1    20       1       0
2    30       1       1
3    31       2       1
4    45       2       1
5    60       2       2

pd.cut documentation
Include parameter right=False

test = pd.DataFrame({'days': [0,31,45]})
test['range'] = pd.cut(test.days, [0,30,60], right=False)

test

   days     range
0     0   [0, 30)
1    31  [30, 60)
2    45  [30, 60)

You can use labels to pd.cut() as well.
The following example contains the grade of students in the range from 0-10.
We’re adding a new column called ‘grade_cat’ to categorize the grades.

bins represent the intervals: 0-4 is one interval, 5-6 is one interval, and so on
The corresponding labels are “poor”, “normal”, etc

bins = [0, 4, 6, 10]
labels = ["poor","normal","excellent"]
student['grade_cat'] = pd.cut(student['grade'], bins=bins, labels=labels)

A sample of how the .cut works

s=pd.Series([168,180,174,190,170,185,179,181,175,169,182,177,180,171])
    pd.cut(s,3)
    #To add labels to bins
    pd.cut(s,3,labels=["Small","Medium","Large"])

This can be used directly on a range

@jezrael has explained almost all the use-cases of pd.cut()

One use-case that i would like to add is the following

pd.cut(np.array([1,2,3,4,5,6]),3)

the number of bins are decided by the second parameter, thus we have following output

[(0.995,2.667],(0.995,2.667],(2.667,4.333],(2.667,4.333], (4.333,6.0], (4.333,6.0]]
Categories (3, interval[float64]): [(0.995,2.667] < (2.667,4.333] < (4.333,6.0]]

Similarly if we use the number of bin parameter(second parameter) as 2 following will be the output

[(0.995, 3.5], (0.995, 3.5], (0.995, 3.5], (3.5, 6.0], (3.5, 6.0], (3.5, 6.0]]
Categories (2, interval[float64]): [(0.995, 3.5] < (3.5, 6.0]]