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If f (x) = [x sin p x] { where [x] denotes greatest integer function}, then f (x) is
- a)Differentiable at x = 1
- b)Continuous in (-1, 0)
- c)Continuous at x = 0
- d)Differentiable in (-1, 1)
Correct answer is option 'C'. Can you explain this answer?
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If f (x) = [x sinpx] { where [x] denotes greatest integer function}, t...
Continuous at x = 0
The greatest integer function [x] means the largest integer less than or equal to x. So, [x sin(xπ)] would mean the greatest integer less than or equal to x sin(xπ).
Understanding the function f(x)
- For x = 0, f(0) = [0 sin(0)] = [0] = 0
- As x approaches 0 from the right, sin(xπ) approaches 0, so f(x) also approaches 0
- As x approaches 0 from the left, sin(xπ) approaches 0, so f(x) also approaches 0
Continuity at x = 0
- Since the left-hand limit and the right-hand limit of f(x) at x = 0 are equal and both equal to f(0), f(x) is continuous at x = 0.
Therefore, the correct answer is option 'C': Continuous at x = 0.
The greatest integer function [x] means the largest integer less than or equal to x. So, [x sin(xπ)] would mean the greatest integer less than or equal to x sin(xπ).
Understanding the function f(x)
- For x = 0, f(0) = [0 sin(0)] = [0] = 0
- As x approaches 0 from the right, sin(xπ) approaches 0, so f(x) also approaches 0
- As x approaches 0 from the left, sin(xπ) approaches 0, so f(x) also approaches 0
Continuity at x = 0
- Since the left-hand limit and the right-hand limit of f(x) at x = 0 are equal and both equal to f(0), f(x) is continuous at x = 0.
Therefore, the correct answer is option 'C': Continuous at x = 0.
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If f (x) = [x sinpx] { where [x] denotes greatest integer function}, then f (x) isa)Differentiable at x = 1b)Continuous in (-1, 0)c)Continuous at x = 0d)Differentiable in (-1, 1)Correct answer is option 'C'. Can you explain this answer?
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