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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
The function $f(x)=[x]^n$ , integer n>=2 (where [y] is the greatest integer less than or equal to y), is discontinuous at all point of
Solution
The function is
\( f(x) = [x]^n , \quad n \geq 2 \)
where \([x]\) is the greatest integer function (GIF).The GIF \([x]\) is discontinuous at all integers. Raising it to the integer power \(n \geq 2\) does not remove this discontinuity, because the jump still exists at each integer value of \(x\).
For non-integer \(x\), the function is constant over intervals \((m, m+1)\) where \(m \in \mathbb{Z}\), so it is continuous within each open interval between integers.
Final Answer: The function is discontinuous at all integers.
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