<p>The value of ∫₋₁⁵ sgn({<i>x</i>}) <i>d</i><i>x</i>, where {⋅} denotes the fractional part function, is:</p>
Step-by-Step Solution
Key Concept: Understand that sgn of fractional part is 0 at integers and 1 elsewhere, but the measure of integer points is zero.
<p>The fractional part function {<i>x</i>} = <i>x</i> − [<i>x</i>] gives values in [0, 1). For <i>x</i> ∈ ℤ, {<i>x</i>} = 0, so sgn({<i>x</i>}) = 0. For non-integer <i>x</i>, {<i>x</i>} > 0, so sgn({<i>x</i>}) = 1. The integral over [−1, 5] accounts for integer discontinuities where the contribution is zero at each integer point.</p>
Correct Answer: D