Continuity and DifferentiabilityHard
Question
The function f(x) = [x]2 -[x2] (where [x] is the greatest integer less than or equal to x), is discontinuous at
Options
A.all integers
B.all integers except o and 1
C.all integers except 0
D.all integers except 1
Solution
Note All integers critical point for greatest integer function.
Case I When x ∈ I
f(x) = [x]2 -[x2] = x2 - x2 = 0
Case I When x ∈ I
If 0 < x < 1, then [x] = 0
and 0 < x2 < 1, then [x2] = 0
Next, if 1 ≤ x2 < 2
⇒ 1 ≤ x < √2
⇒ [x] =1 and [x2] = 1
Therefore, f(x) = [x]2 -[x2] = 0, if 1 ≤ x < √2
Therefore, f(x) = 0, if 0 ≤ x < √2
This shows thaty f (x) is continuous at x = 1
Therefore, f(x) is discontinuous in (- ∞, 0)
[√2, ∞) on many other points.
Therefore, (b) is the answer.
Case I When x ∈ I
f(x) = [x]2 -[x2] = x2 - x2 = 0
Case I When x ∈ I
If 0 < x < 1, then [x] = 0
and 0 < x2 < 1, then [x2] = 0
Next, if 1 ≤ x2 < 2
⇒ 1 ≤ x < √2
⇒ [x] =1 and [x2] = 1
Therefore, f(x) = [x]2 -[x2] = 0, if 1 ≤ x < √2
Therefore, f(x) = 0, if 0 ≤ x < √2
This shows thaty f (x) is continuous at x = 1
Therefore, f(x) is discontinuous in (- ∞, 0)
[√2, ∞) on many other points.Therefore, (b) is the answer.
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