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Question #41307

Difficulty: 8/10 | 🟢 Free

The Question

Consider the function $f(x) = \\left[ \\frac{6}{\\sqrt{x^2 - 2x} + 1} \\right]$ , where $[ \\cdot ]$ denotes the greatest integer function. Let $D$ be the domain of $f(x)$ . Find the number of points of discontinuity of $f(x)$ in the set $D \\cap [-3, 5]$ .

Options

A.
8
B.
9
C.
10
D.
12

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Composition Continuity|Domain Restrictions|Greatest Integer Function (GIF)

Why This Question Is Tricky

Common Mistakes

This question tests the 'double-barrier' continuity trap involving composite functions with domain restrictions and the Greatest Integer Function (GIF).

Students typically find the points where the inner function equals an integer, correctly identifying the 8 internal jump discontinuities.

However, the lethal trap lies at the boundary points of the domain restriction ( $x=0$ and $x=2$ ).

Because the inner function approaches an integer (6) from below on the valid side of the domain, the one-sided limit of the GIF drops to 5, creating a discontinuity at the boundary itself.

Students who only check for integer crossings and assume continuity at closed boundaries will incorrectly answer 8.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Composition Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free