Generate random number from custom distribution

I am trying to generate random numbers from a custom distribution, i already found this question: Simulate from an (arbitrary) continuous probability distribution but unfortunatly it does not help me since the approach suggested there requires a formula for the distribution function. My distribution is a combination of multiple uniform distributions, basically the distribution function looks like a histogram. An example would be:

f(x) = { 
    0     for  x < 1
    0.5   for  1 <= x < 2
    0.25  for  2 <= x < 4
    0     for  4 <= x
}