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

We are given \(f:(0,1) \to \{0,1\}\) where \(f(r)=1\) if the second digit after the decimal point of \(r\) belongs to \{2,3,6,7\}, and \(f(r)=0\) otherwise.

Step 1: Divide \((0,1)\) into 100 half open strips \(I_k = [k/100, (k+1)/100)\) for \(k=0,1,\dots,99\). For any \(r \in I_k\), the second digit after the decimal point equals \(k \bmod 10\), and this digit is fixed throughout the strip.

Step 2: Since \(f\) depends only on \(k \bmod 10\), the value of \(f\) on strip \(I_k\) repeats with period 10 in \(k\). For \(k \bmod 10 = 0,1,2,\dots,9\) the values are: \[0,0,1,1,0,0,1,1,0,0\] since the digit is one of 2, 3, 6, 7 exactly when \(k \bmod 10\) takes those values.

Step 3: Because \(f\) is constant on every open strip, discontinuities can only occur at the 99 strip boundaries \(r=k/100\), \(k=1,\dots,99\). At such a boundary the left hand value equals the constant on strip \(k-1\), while the value at and just right of the point equals the constant on strip \(k\). So \(r=k/100\) is a discontinuity exactly when the pattern value changes from position \((k-1)\bmod 10\) to position \(k \bmod 10\).

Step 4: Within one period \(0,0,1,1,0,0,1,1,0,0\), compare consecutive entries: 0-0 same, 0-1 change, 1-1 same, 1-0 change, 0-0 same, 0-1 change, 1-1 same, 1-0 change, 0-0 same. This gives exactly 4 changes inside each period of 10. The pattern repeats 10 times as \(k\) runs 0 to 99 (blocks 0-9, 10-19, ..., 90-99), giving \(4 \times 10 = 40\) internal changes. At the 9 junctions joining consecutive blocks, the last value (0) equals the first value (0) of the next block, so no extra change occurs there.

Step 5: Hence the total number of points in \((0,1)\) at which \(f\) is discontinuous equals the number of pattern changes found above.

Answer: 40