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Let I = (1) u (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.?
Most Upvoted Answer
Let I = (1) u (2) R.. For x ∈ R, let @(x) = dist(x, I) = inf {x-y: y €...
Understanding the Function @ and the Set I
The given set I = (1) ∪ (2) R consists of the point 1 and the entire set of real numbers greater than 2. The function @(x) represents the distance from x to the closest point in I.
Behavior of the Function @
- For x < 1:="" />
- Distances are calculated to point 1, thus, @(x) = 1 - x.
- For 1 ≤ x < />
- The nearest point is 1, so @(x) = x - 1.
- For x = 2:
- Here, @(2) = 1 (distance to point 1).
- For x > 2:
- The nearest point is 2, thus, @(x) = x - 2.
Continuity of @
- The function @(x) is continuous on R.
- At transition points (1 and 2), the limits from both sides equal the value of the function at those points.
Differentiability of @
- At x = 1:
- The left-hand derivative is -1, and the right-hand derivative is +1, hence not differentiable.
- At x = 2:
- Similarly, the left-hand derivative is +1, while the right-hand derivative is 0, indicating non-differentiability.
Conclusion
- The function @ is continuous everywhere on R.
- It is not differentiable only at x = 1 and x = 2.
Thus, the correct answer is (c): @ is continuous on R but not differentiable only at x = 1 and 2.
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Let I = (1) u (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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Let I = (1) u (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.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Let I = (1) u (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.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I = (1) u (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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