How to drop connecting lines where the function is discontinuous

use nan to break the line into multiple segments:

import numpy as np
from pylab import *
x=linspace(0,1,100)    
y=zeros(100)
y[x<0.5] = x[x<0.5]
y[x>=0.5] = 1 + x[x>=0.5]

pos = np.where(np.abs(np.diff(y)) >= 0.5)[0]

x[pos] = np.nan
y[pos] = np.nan

plot(x, y, '-o')

Edit:

to insert nan at discontinuities:

pos = np.where(np.abs(np.diff(y)) >= 0.5)[0]+1
x = np.insert(x, pos, np.nan)
y = np.insert(y, pos, np.nan)