Question:
Let $ [t] $ represent the greatest integer not exceeding $ t $. The number of discontinuous points of $ [10^t] $ in $ (0, 10) $ is:
Let $ [t] $ represent the greatest integer not exceeding $ t $. The number of discontinuous points of $ [10^t] $ in $ (0, 10) $ is:
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The greatest integer function \( [x] \) is discontinuous at every integer \( x \). Solve for when the argument is an integer to find discontinuity points.
Updated On: Mar 18, 2026
- \( 10^{10} - 1 \)
- \( 10^{10} \)
- \( 10^{10} - 2 \)
- \( e^{10} \)
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The Correct Option is C
Solution and Explanation
Step 1: Understand the greatest integer function.
\( [10^t] \) is discontinuous when \( 10^t \) is an integer, as the floor function jumps at integers.
Step 2: Determine the range of \( 10^t \).
For \( t \in (0, 10) \), \( 10^t \) ranges from just above 1 to \( 10^{10} \).
Step 3: Find the discontinuities.
Discontinuity at \( 10^t = n \): \( t = \log_{10} n \). Require \( 0<t<10 \):
\[ 1 \leq n<10^{10}. \] Discontinuities at \( n = 2 \) to \( 10^{10} - 1 \), totaling \( 10^{10} - 2 \).
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