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Question #40389

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \frac{x^2}{196}$ and $g(x) = [x]$ , where $[ \cdot ]$ denotes the greatest integer function. Find the number of points of discontinuity of the function $h(x) = g(f(x)) - f(g(x))$ in the interval $(0, 196)$ .

Options

A.
377
B.
390
C.
364
D.
378

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

Greatest Integer Function (GIF)|Composition Continuity|Composition Non-Commutativity

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of the Greatest Integer Function under composition, specifically analyzing the continuity of $f([x])$ versus $[f(x)]$ .

The primary trap is assuming that at points where both inner and outer functions have discontinuities, the composite difference might become continuous, or conversely, failing to account for overlapping jump points.

Students often blindly add the number of discontinuities ( $195 + 195 = 390$ ) without using the Principle of Inclusion-Exclusion.

Furthermore, it forces a rigorous left-hand and right-hand limit check at the 13 intersection points to prove that the jumps do not cancel each other out, punishing those who rely on visual intuition rather than formal limit definitions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Greatest Integer Function (GIF)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free