A function Φ of the reals is called a step function on [a,b] iff:

For some finite set of points

a = b0 < b1 < ... < bn = b

Φ is constant on the interval (bi-1 , bi) for i = 1,2,3, ... , n. In other words, Φ looks like this:
            |
    **      |** 
       **   |  
  **      **|  **
            |
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The integral of a step function is given by:

∫ Φ(x)dx = Σ (bi - bi)ci