UPSC Exam > UPSC Questions > Let I = (1) (2) R.. For x ∈ R, let @(x) = dis...
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Let I = (1) (2) R.. For x ∈ R, let @(x) = dist(x, I) = inf {x-y: y € I}. Then
(a) @ is continuous on R but not differentiable only at x = 1, 3/2 and 2
(b) @ is continuous on R but not differentiable only at x = 1
(c) @s continuous on R but not differentiable only at x = 1 and 2.
(d) @ is discontinuous somewhere on R.?
(a) @ is continuous on R but not differentiable only at x = 1, 3/2 and 2
(b) @ is continuous on R but not differentiable only at x = 1
(c) @s continuous on R but not differentiable only at x = 1 and 2.
(d) @ is discontinuous somewhere on R.?
Most Upvoted Answer
Let I = (1) (2) R.. For x ∈ R, let @(x) = dist(x, I) = inf {x-y: y € I...
Understanding the Function @(x)
The function @(x) = dist(x, I) denotes the distance from a point x in R to the set I = (1, 2). Here's a detailed breakdown of its properties.
Continuity of @(x)
- @(x) is continuous on R:
- The distance function to a closed interval is continuous. Points outside the interval (1, 2) will have distances increasing or decreasing smoothly as they move away from the interval.
Points of Non-Differentiability
- @(x) is not differentiable at specific points:
- The function may experience sharp changes in slope at the boundaries of the interval (1, 2).
- Specifically, @(x) is not differentiable at:
- x = 1: The function transitions from decreasing to increasing.
- x = 2: The function also changes from decreasing to increasing.
- Consideration of x = 3/2 shows that it does not have a sharp transition, so it is differentiable there.
Conclusion
- Based on the analysis:
- @(x) is continuous everywhere on R.
- The function is not differentiable only at x = 1 and x = 2.
- Therefore, the correct choice is:
- (c) @(x) is continuous on R but not differentiable only at x = 1 and 2.
This conclusion emphasizes the behavior of the function around the interval's endpoints and confirms the overall continuity across the real line.
The function @(x) = dist(x, I) denotes the distance from a point x in R to the set I = (1, 2). Here's a detailed breakdown of its properties.
Continuity of @(x)
- @(x) is continuous on R:
- The distance function to a closed interval is continuous. Points outside the interval (1, 2) will have distances increasing or decreasing smoothly as they move away from the interval.
Points of Non-Differentiability
- @(x) is not differentiable at specific points:
- The function may experience sharp changes in slope at the boundaries of the interval (1, 2).
- Specifically, @(x) is not differentiable at:
- x = 1: The function transitions from decreasing to increasing.
- x = 2: The function also changes from decreasing to increasing.
- Consideration of x = 3/2 shows that it does not have a sharp transition, so it is differentiable there.
Conclusion
- Based on the analysis:
- @(x) is continuous everywhere on R.
- The function is not differentiable only at x = 1 and x = 2.
- Therefore, the correct choice is:
- (c) @(x) is continuous on R but not differentiable only at x = 1 and 2.
This conclusion emphasizes the behavior of the function around the interval's endpoints and confirms the overall continuity across the real line.
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Let I = (1) (2) R.. For x ∈ R, let @(x) = dist(x, I) = inf {x-y: y € I}. Then(a) @ is continuous on R but not differentiable only at x = 1, 3/2 and 2(b) @ is continuous on R but not differentiable only at x = 1(c) @s continuous on R but not differentiable only at x = 1 and 2.(d) @ is discontinuous somewhere on R.?
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