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.

The answer is 0.5.

1 solution

Ivan Barreto
Apr 27, 2017

We may see $h$ as the uniform limit of functions $h_{2n}$ , where $h_{2n}$ only operates on the first $2n$ decimal digits, and leaves the rest unchanged. Since each $h_{2n}$ applies a bijection to the first $2n$ digits, we can subdivide the domain in intevals of size $10^{-2n}$ and reorder it back into the identity function. This shows that the integral of each $h_{2n}$ is $\frac{1}{2}$ . Uniform convergence then shows that the limiting function also has integral of $\frac{1}{2}$ , so the answer is: $\fbox{0.5}$

Digital integration

The answer is 0.5.

1 solution

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