Problem of the Week 1238

Change of Base

Define f from [0, 1] to [0, 1] by taking a real x, writing it in base 2, and then reinterpreting that expression in base 10.

For example, f(1/3) = f(0.010101...) = 1/99.

What is the integral of f from 0 to 1?

You may assume that f is Riemann integrable (this is so because there are countably many discontinuities).

Source: The problem is Problem 6 on the 2014 University of Illinois Math Contest. Heard from Keith Kearnes, Univ. of Colorado.

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