Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The set of points of discontinuities of the function \(f(x)=\sqrt{x}-[\sqrt{x}]\), where \([x]\) denotes the greatest inte- ger less than or equal to \(x\), contains the set (A) \((-\infty, 0)\) (B) \(\left\\{n^{2}: n \in N\right\\}\) (C) \(N\) (D) \(\\{2 n: n \in N\\}\)

Step-by-Step Solution

Answer

The points of discontinuity are \( \{n^2 : n \in N\} \), so option (B).

Step 1: Understand the Function

The function given is \(f(x) = \sqrt{x} - [\sqrt{x}]\). Here, \([x]\) denotes the greatest integer less than or equal to \(x\). This function primarily takes the fractional part of \(\sqrt{x}\) and is written as \(f(x) = x - [x]\), when substituted with \(\sqrt{x}\).

Step 2: Identify Breakpoints of Discontinuity

The function \(\sqrt{x} - [\sqrt{x}]\) is discontinuous at points where \(\sqrt{x}\) is an integer because, at these values, the fractional part becomes zero. This happens when \(x = n^2\), where \(n\) is an integer since \(\sqrt{x} = n\) results in an integer.

Step 3: Analyze the Given Options

We need to evaluate which set corresponds to the integers where \(\sqrt{x}\) is an integer. Option (B) \(\{n^2 : n \in N\}\) identifies the points where \(x = n^2\), which are the discontinuities because at these points, \(f(x)\) jumps from near 1 (as \(x\) approaches \(n^2\) from the left) to 0.

Step 4: Verify Other Options

Option (A) \((-ifty, 0)\) is incorrect as \(f(x)\) is not defined for negative values. Option (C) \(N\) is incorrect as not all natural numbers are perfect squares. Option (D) \(\{2n : n \in N\}\) is incorrect since not all even numbers are perfect squares.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([x]\), is a mathematical operation that takes any real number \(x\) and assigns it the largest integer less than or equal to \(x\). This function is also commonly known as the floor function.

• This function will output an integer value.
• For positive non-whole numbers, it rounds down to the nearest lower whole number.
• For negative numbers, it moves toward negative infinity.

Consider \(x = 3.7\), therefore \([3.7] = 3\). For a negative example, \(x = -2.3\), then \([-2.3] = -3\). This function is crucial when determining points of discontinuity in functions like the one discussed here, as it provides the integer base used in further calculations.

Fractional Part Function

The fractional part function, denoted usually as \(\{x\}\), represents the difference between a real number \(x\) and the greatest integer less than or equal to \(x\). Essentially, it extracts the decimal component of the number.

• Mathematically, \(\{x\} = x - [x]\).
• The function returns a value in the range \(0 \leq \{x\} < 1\).

For instance, with a number \(x = 5.6\), \([5.6] = 5\) hence \(\{5.6\} = 5.6 - 5 = 0.6\). The fractional part function is uniquely useful in understanding the continuity of functions, especially when combined with other elements as discussed in the exercise.

Perfect Squares

A perfect square is an integer that can be expressed as the square of another integer. These numbers take the form \(n^{2}\), where \(n\) is an integer. Common examples include \(1, 4, 9, 16,\) and so on.

• They occur naturally when an integer is multiplied by itself.
• Perfect squares are integral in identifying discontinuities in functions involving the floor or greatest integer functions, particularly when the function involves square roots.

In our example function \(f(x) = \sqrt{x} - [\sqrt{x}]\), discontinuities occur precisely when \(x\) is a perfect square, because at these points, the fractional part shifts abruptly from near 1 back to 0.

JEE Main Mathematics

The Joint Entrance Examination (JEE) Main is a standardized test in India for students aspiring to enter engineering colleges. Mathematics is one of the primary subjects tested, and it includes topics ranging from basic algebra to more complex concepts such as calculus and number theory.

• Understanding functions and their properties, including points of discontinuity, is essential.
• Students are tested on their ability to manipulate and analyze functions involving the greatest integer and fractional part functions.
• Key problems often include determining points, plotting functions, and understanding mathematical behavior.

The problem discussed, involving discontinuities and perfect squares, is reflective of typical JEE Main mathematics questions in its requirement for deep conceptual understanding and application of learned mathematical principles.