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piecewise definition of a function

I'd like to define a function in a piecewise manner.  As of yet I've 
been unsuccessful with even the simplest example.  Consider the 

first define a 1-d bump,

f[x_] = If[x < 0, 0,If[x > 1, 0, 1]]
  Mathematica doesn't seem to mind this, as f[0.1] evaluates 1 and 
f[-23] evaluates 0.
The problem comes when I try to integrate the function.  The 
following command which should evaluate to 1 doesn't work AT ALL!

Integrate[f[x], {x, -50, 50}]

What is the proper technique here?

Nathan Moore
University of Minnesota Physics

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