#!/usr/bin/env python
# coding: utf-8

# CDF:
# $$
# F(x) = \begin{cases}
# 0 & x <0,\\
# x & 0 \leq x < 0.5,\\
# \frac{x}{2} + 0.5 & 0.5 \leq x \leq 1\\
# 1 & x>2
# \end{cases}
# $$
# 
# P*D*F:
# 
# $$
# f(x) = \begin{cases}
# 0 & x <0,\\
# 1 & 0 \leq x < 0.5,\\
# \frac{1}{2}  & 0.5 \leq x \leq 1\\
# 0 & x>2
# \end{cases}
# $$
# 
# $\int f(t) dt = 0.75$??
# 
# What's missing?
# 
# p[X=0.5] = 0.25. But defining such a probability should be $0$ for a P*D*F.[It's technically a 'PMF'. The mass at 0.5 being 1/4]

# In[15]:


get_ipython().run_line_magic('matplotlib', 'inline')
from __future__ import division
def F(x):
    if x<0:
        return 0
    if x<0.5:
        return x
    if x>=0.5 and x<=1:
        return x/2+0.5
    return 1
    
import numpy as np
import matplotlib.pyplot as plt
x = np.linspace(-2,2, num=100)
fx =[F(i) for i in x]
pos = np.where((x<=0.51) & (x>=0.49))[0]

x[pos] = np.nan
fx[pos] = np.nan
plt.plot(x,fx)