Cubic spline interpolation factors required to pass through original datapoints (scipy, python)

I am using scipy.interpolate.interp1d for cubic spline interpolation of a signal. While I believe interpolated signal should pass through all original data points, this is not the case when interpolating with certain factors.

e.g. if there are N samples, with N-1 spaces between samples and an interpolation factor of f, we can insert x points between the samples N*f == (N-1)*x + N. If x is not a whole number, the interpolated signal cannot pass through the original data points. As expected this is the case, code using scipy below with N = 4 and interpolation factor f of 3 or 4.

My question is A) is this correct or am I doing something wrong? and B) Is the formula above where x is a whole number a sufficient check that the original data samples will appear in the interpolated signal (or maybe there are edge cases).

Many Thanks

import scipy.interpolate
import numpy as np

# produce random data and interp
x = np.linspace(0, 2, 4)
np.random.seed(123)
y = np.random.random(4)

interp_f = scipy.interpolate.interp1d(x, y, kind='cubic')

# upsample factor 4
x_f4 = np.linspace(0, 2, 16)
y_f4 = interp_f(x_f4)

# upsample factor 3
x_f3 = np.linspace(0, 2, 12)
y_f3 = interp_f(x_f3)

print("Sample 2 in raw data: {0:.10f}, Sample 6 in interp f4: {1:.10f}, Sample 4 in interp f3: {2:.10f}".format(y[1], y_f4[5], y_f3[4]))
# Sample 2 in raw data: 0.2861393350, Sample 6 in interp f4: 0.2861393350, Sample 5 in interp f3: 0.2657521625

from Cubic spline interpolation factors required to pass through original datapoints (scipy, python)