Digital integration

Calculus Level pending

Define the function \(f(x)\) on \((0,1)\) as the one that takes the standard decimal representation of \(x\) and swaps all odd and even indices, and then interprets the resulting sequence of digits as the decimal representation of another real number: \[f(0.x_1x_2x_3x_4\ldots) = 0.x_2x_1x_4x_3\ldots.\] Now define \(g(x)\) on \((0,1)\) as a function that takes each digit \(x_n\) in the decimal representation of \(x\) and replaces it with \(7x_n + 2 \pmod{10},\) and then similarly interprets the result as a new real number.

Find the integral of \(h(x)=g\big(f(x)\big)\) over its entire domain.

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