Question

Let $ [t] $ represent the greatest integer not exceeding $ t $. The number of discontinuous points of $ [10^t] $ in $ (0, 10) $ is:

• A. \( 10^{10} - 1 \)
• B. \( 10^{10} \)
• C. \( 10^{10} - 2 \)
• D. \( e^{10} \)

Hint:

The greatest integer function \( [x] \) is discontinuous at every integer \( x \). Solve for when the argument is an integer to find discontinuity points.

Answer 1

The Correct Option is C

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 \( 10^{10} - 2 \).