Question

Find number of points of discontinuity of the function \( f(x) = [x^2 - x + 2] \) in \( x \in [2,4] \) (where \( [\ ] \) denotes greatest integer function).

• A. 9
• B. 11
• C. 8
• D. 10

Hint:

For \( [g(x)] \), discontinuities occur where \( g(x) \) becomes an integer. First find the range of \( g(x) \), then count the integer values in that range carefully, keeping endpoint treatment in mind.

Answer 1

The Correct Option is D

Step 1: Understand when \( [g(x)] \) is discontinuous.
The greatest integer function \( [g(x)] \) is discontinuous at those points where \( g(x) \) takes an integer value.
Here,
\[ g(x)=x^2-x+2. \] So, we need to find how many values of \( x \in [2,4] \) make \( x^2-x+2 \) an integer. More precisely, since \( g(x) \) is continuous and strictly increasing on \( [2,4] \), there will be one discontinuity for each integer value attained by \( g(x) \).

Step 2: Find the range of \( g(x) \) on \( [2,4] \).
Now evaluate \( g(x) \) at the endpoints:
\[ g(B)=2^2-2+2=4, \] \[ g(D)=4^2-4+2=14. \] Also,
\[ g'(x)=2x-1. \] For \( x \in [2,4] \), we have \( 2x-1>0 \), so \( g(x) \) is strictly increasing on \( [2,4] \).
Hence the range of \( g(x) \) on \( [2,4] \) is
\[ [4,14]. \]
Step 3: Count the integer values in this range.
The integers from \( 4 \) to \( 14 \) are
\[ 4,5,6,7,8,9,10,11,12,13,14. \] Total number of integers is
\[ 14-4+1=11. \]
Step 4: Check endpoint behavior.
At \( x=2 \),
\[ g(B)=4, \] which is an integer, but since \( x=2 \) is the left endpoint of the interval, it does not create a discontinuity in the interior in the usual counting for such questions.
At \( x=4 \),
\[ g(D)=14, \] which is also an endpoint value.
Thus, excluding one endpoint jump in the standard school-level counting used here, the number of discontinuity points comes out to be
\[ 11-1=10. \]
Final Answer: \( 10 \)