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What is the point of discontinuity for signum function?
- a)x=1
- b)x=-1
- c)x=0
- d)function is continuous on R
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
What is the point of discontinuity for signum function?a)x=1b)x=-1c)x=...
It has a jumped discontinuity which means if the function is assigned some value at the point of discontinuity it cannot be made continuous. But the function is definitely discontinuous at x=0. the sgn function is a discontinuous function (isolated/jump discontinuity).
Community Answer
What is the point of discontinuity for signum function?a)x=1b)x=-1c)x=...
Explanation:
The signum function is defined as follows:
- sgn(x) = -1 for x < />
- sgn(x) = 0 for x = 0
- sgn(x) = 1 for x > 0
This function is discontinuous at x = 0.
Reason:
The reason why the signum function is discontinuous at x = 0 is because the limit of the function as x approaches 0 from the left is -1, while the limit of the function as x approaches 0 from the right is 1. Since these two limits are not equal, the function is discontinuous at x = 0.
To demonstrate this, we can use the definition of a limit:
- lim x->0- sgn(x) = lim x->0- (-1) = -1
- lim x->0+ sgn(x) = lim x->0+ (1) = 1
Since the limits from the left and right are not equal, the limit does not exist at x = 0, and the function is discontinuous at this point.
Conclusion:
Therefore, the correct answer is option 'C'. The signum function is discontinuous at x = 0 because the limits from the left and right are not equal.
The signum function is defined as follows:
- sgn(x) = -1 for x < />
- sgn(x) = 0 for x = 0
- sgn(x) = 1 for x > 0
This function is discontinuous at x = 0.
Reason:
The reason why the signum function is discontinuous at x = 0 is because the limit of the function as x approaches 0 from the left is -1, while the limit of the function as x approaches 0 from the right is 1. Since these two limits are not equal, the function is discontinuous at x = 0.
To demonstrate this, we can use the definition of a limit:
- lim x->0- sgn(x) = lim x->0- (-1) = -1
- lim x->0+ sgn(x) = lim x->0+ (1) = 1
Since the limits from the left and right are not equal, the limit does not exist at x = 0, and the function is discontinuous at this point.
Conclusion:
Therefore, the correct answer is option 'C'. The signum function is discontinuous at x = 0 because the limits from the left and right are not equal.
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