Consider the following statements: I. f(x) is continuous at x=0. II. f(x) is continuous at x=1 Which of the statements given above is/are correct?
Direction: Consider the following for the two (02) items that follow : Let f(x)=[x]^{2}-[x^{2}].
- A. I only
- B. II only ✓
- C. Both I and II
- D. Neither I nor II
Correct Answer: B. II only
Explanation
At x=0, f(0) = 0. RHL \lim_{x \to 0^+} f(x) = 0^2 - 0 = 0. LHL \lim_{x \to 0^-} f(x) = (-1)^2 - 0 = 1. LHL \neq RHL, so it's discontinuous at x=0. At x=1, f(1) = 1^2 - 1 = 0. RHL \lim_{x \to 1^+} f(x) = 1^2 - 1 = 0. LHL \lim_{x \to 1^-} f(x) = 0^2 - 0 = 0. Since LHL = RHL = f(1), f(x) is continuous at x=1.
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