The Cantor Function

The following images represent a progressive construction of the Cantor function and were created by Mathematica.

The Cantor function is a continuous increasing function, f(x), for which f(0) = 0 and f(1) = 1.  In addition it is a singular function, a function whose derivative is  0  almost everywhere.  Finally it represents a function for which the Lebesgue Integral of  f ' (x)  on [0, 1] has the value of  0, but f(1) - f(0) = 1. Thus the Lebesgue integral of f ' (x)  on [0, 1] does not equal f(1) - f(0) .

If you would like to see an animated construction of the Cantor function, click here .