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

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 correct option is (B) \( \{n^2: n \in N\} \).

Step 1: Understanding the Function Components

The function \( f(x) = \sqrt{x} - [\sqrt{x}] \) is composed of two parts: \( \sqrt{x} \), which is a real-valued function defined for \( x \geq 0 \), and \( [\sqrt{x}] \), which is a step function representing the greatest integer less or equal to \( \sqrt{x} \). The function \( f(x) \) represents the fractional part of \( \sqrt{x} \) as \( \sqrt{x} \) will always be expressed as an integer plus a fractional component.

Step 2: Analyzing Discontinuity

Discontinuities arise when \( f(x) \) jumps abruptly. Since \( f(x) = \sqrt{x} - [\sqrt{x}] \) represents the fractional part of \( \sqrt{x} \), it ranges from 0 (inclusive) to just below 1. The function 'jumps' to 0 whenever \( \sqrt{x} \) is a whole number.

Step 3: Identifying Specific Points

Points where \( \sqrt{x} \) is an integer are the exact discontinuities. Solving \( \sqrt{x} = n \) where \( n \) is an integer, we find \( x = n^2 \). This means the function jumps at each perfect square. Thus, \( \{n^2: n \in N\} \) is the set of discontinuity points.

Step 4: Conclusion Based on Options

The description of discontinuities aligns with the set of perfect squares, \( \{n^2: n \in N\} \). Therefore, option (B) is correct.

Key Concepts

Greatest Integer Function

The greatest integer function, denoted as \([x]\), is a type of step function that rounds down any number to the nearest integer less than or equal to it. For instance, if you have a number like 3.7, the greatest integer function \([3.7]\) would be 3. The function essentially strips away the decimal part of any real number. This feature makes it inherently discontinuous at every integer point, because as you approach an integer from the left, the function outputs the integer value, but suddenly at the integer, the output remains the same, causing a jump in the function's graph.

• The function \([x]\) is constant on the interval from \(n\) to \(n+1\) for any integer \(n\).
• At each integer \(n\), there is a jump discontinuity.
• This characteristic is useful for creating a variety of interesting mathematical functions, like the one in this problem.

Understanding this function is key, especially when analyzing functions that involve both continuous and discontinuous behavior together.

Perfect Squares

Perfect squares are numbers that can be expressed as the product of an integer multiplied by itself. Examples include 1, 4, 9, 16, and so on, corresponding to \(1^2, 2^2, 3^2,\) and so forth. They are crucial when exploring the problem of discontinuity in functions, particularly those involving square roots, as they represent key points where function behavior might change.

• At a perfect square, \(x = n^2\), the square root yields an integer \(n\).
• In the given function \(f(x) = \sqrt{x} - [\sqrt{x}]\), perfect squares are the points where \(\sqrt{x}\) becomes exactly an integer, leading to a complete reset of the fractional part to zero, hence, a discontinuity.
• Understanding when these occur (for instance, by recognizing that the square root of a perfect square is always an integer) helps identify the discontinuous nature of some functions.

Functions and Graphs

Understanding functions and their graphs is foundational in visualizing mathematical relationships. Functions define a relationship between inputs and outputs, offering a visual method to explore these associations through graphs. Each function has unique characteristics based on its components, making graphical analysis an indispensable tool.Consider the problem with the function \(f(x) = \sqrt{x} - [\sqrt{x}]\):

• The function \(\sqrt{x}\) is continuous and smooth for all \(x\geq 0\), displaying a gentle curve upwards.
• The greatest integer component \([\sqrt{x}]\) behaves like a staircase, remaining constant until specific points, then jumping abruptly, depicting its discontinuous nature at integer points.
• Combining these two results in a function that generally stays below 1 but resets to 0 at integers, creating a characteristic sawtooth pattern.
• Visualizing helps clarify why discontinuities appear at certain points—where the fractional part resets to zero at perfect squares.

Graphical representation in tandem with analytical exploration allows for a deeper understanding of how discontinuities form and where they occur.