Question

For x ∈ ℝ, let ⌊x⌋ denote the greatest integer less than or equal to x.
For x, y ∈ ℝ, define
\(\min\left\{x,y\right\} = \begin{cases}     x & \text{if } x \le y, \\     y & \text{otherwise.} \end{cases}\)
Let f:[−2𝜋, 2𝜋] → ℝ be defined by
f(x) = sin(min{x, x − ⌊x⌋}) for x ∈ [−2𝜋, 2𝜋].
Consider the set S = {x ∈ [−2𝜋, 2𝜋]: f is discontinuous at x}.
Which one of the following statements is TRUE ?

• A. S has 13 elements
• B. S has 7 elements
• C. S is an infinite set
• D. S has 6 elements

Answer 1

The Correct Option is D

- The function \( f(x) = \sin(\min\{x, x - |x|\}) \) is discontinuous where the argument inside the sine function changes its value abruptly. The function \( \min\{x, x - |x|\} \) will change its value at the integer points because \( |x| \) changes at integer values, creating potential discontinuities at such points. - The set \( S \) consists of the points where this discontinuity occurs. These points are the integer multiples of \( \pi \) within the interval \( [-2\pi, 2\pi] \). - The multiples of \( \pi \) in this interval are \( -2\pi, -\pi, 0, \pi, 2\pi \), which gives 6 distinct points where \( f(x) \) is discontinuous. Thus, the set \( S \) has exactly 6 elements. Therefore, the correct answer is (D): \( S \) has 6 elements.