Continuity
Let f(x)=[x]2+{x}, where [.] denotes the greatest integer function and {.} denotes the fractional part. Discuss the continuity of f(x) at integer points x=n.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- continuity
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
Continuous only at x=1
We analyze the continuity at an integer n.
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Right Hand Limit (RHL) at x→n+: [x]=n,{x}→0. So, RHL =n2+0=n2.
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Left Hand Limit (LHL) at x→n−: [x]=n−1,{x}→1. So, LHL =(n−1)2+1=(n−1)2+1.
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Condition for continuity: RHL = LHL ⟹n2=(n−1)2+1 ⟹n2=n2−2n+1+1 ⟹0=−2n+2 ⟹2n=2⟹n=1.
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Verification at n=1: f(1)=12+0=1. RHL =1. LHL =02+1=1. Since LHL = RHL =f(1), f(x) is continuous at x=1. For all other integers, it is discontinuous.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of continuity. It appeared in the 2025 exam.
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