If f: R → R is a function defined by f(x) = [x] ≤ft( 2x - 1/2 π ) , where [x] denotes the greatest integer function, then f is: - Mathematics MCQ AIEEE 2012 2026 | ExamTest.live

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If f : R \to R is a function defined by f (x) = [x] cos (\frac{2 x - 1}{2} π), where [x] denotes the greatest integer function, then f is:

Solution & Step-by-step Explanation

f (x) = [x] cos (x π - \frac{π}{2}) = [x] sin (x π) .The function sin (x π) is continuous everywhere. The term [x] is discontinuous at all integers n .At an integer x = n : LHL = lim_{h \to 0} [n - h] sin ((n - h) π) = (n - 1) sin (nπ) = (n - 1) \cdot 0 = 0 RHL = lim_{h \to 0} [n + h] sin ((n + h) π) = n sin (nπ) = n \cdot 0 = 0 f (n) = [n] sin (nπ) = n \cdot 0 = 0 .Since LHL = RHL = f (n) for all integers n, and the function is continuous for non-integers, f (x) is continuous for all x \in R .

Try it yourself before checking the explanation above.

If f : R \to R is a function defined by f (x) = [x] cos (\frac{2 x - 1}{2} π), where [x] denotes the greatest integer function, then f is:

continuous for every real x

discontinuous only at x = 0

discontinuous only at non-zero integral values of x

continuous only at x = 0

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