Question

Consider a function 𝑓: (0,1) →{0, 1} defined as follows.
For a real number 𝑟∈(0,1) , 𝑓(𝑟) = 1 if the second digit after the decimal point
in 𝑟 is one of the four digits 2, 3, 6 and 7. Otherwise, 𝑓(𝑟) is equal to 0.
The number of points in (0,1) at which 𝑓 is discontinuous is ___________. (answer
in integer)

Hint:

Split (0,1) into 100 strips of width 0.01 based on the second decimal digit; f is constant on each strip, and it can only jump at strip boundaries where the repeating period-10 pattern 0,0,1,1,0,0,1,1,0,0 changes value.

Answer 1

Correct Answer: 40

Here is the same result reached by tracking indicator flips strip by strip instead of listing the whole pattern at once.

Write \(r = 0.a_1 a_2 a_3 \ldots\) Every point of \((0,1)\) lies in exactly one strip \(S_k = [\frac{k}{100}, \frac{k+1}{100})\) for some \(k \in \{0,1,\ldots,99\}\), and inside \(S_k\) the second decimal digit \(a_2\) is fixed and equals \(k \bmod 10\).

Define \(g(k) = 1\) if \(k \bmod 10 \in \{2,3,6,7\}\) and \(g(k) = 0\) otherwise; this is exactly the constant value \(f\) takes on strip \(S_k\).

Since \(f\) is constant on the interior of each strip, it can only jump at the 99 strip boundaries \(\frac{1}{100}, \frac{2}{100}, \ldots, \frac{99}{100}\), and a jump occurs at \(\frac{k}{100}\) exactly when \(g(k-1) \ne g(k)\).

Listing \(g(0), g(1), \ldots, g(9) = 0,0,1,1,0,0,1,1,0,0\), and noting \(g\) is periodic with period 10, this exact block of 10 values repeats for \(k=10,\ldots,19\), then \(20,\ldots,29\), and so on, ten times up to \(k=99\).

Inside each block of 10 the sequence flips value exactly 4 times, at the steps where the digit pattern moves from 0 to 1, then 1 to 0, then 0 to 1, then 1 to 0 again. The pair joining the last value of one block (0) to the first value of the next block (0) never flips, so the 9 junctions between blocks contribute nothing.

With 10 blocks each contributing 4 internal flips and zero flips at the 9 junctions, the total count of jump points is \(10 \times 4 = 40\).

So f has exactly 40 points of discontinuity in \((0,1)\).

Final answer: \(40\)