Problem 10 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=[x]^{2}-\left[x^{2}\right]\) (where \([x]\) is the greatest integer less than or equal to \(x\) ), is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1

Step-by-Step Solution

Answer

(B) all integers except 0 and 1

Step 1: Understand the Function Components

The function given is \( f(x) = [x]^2 - [x^2] \). Here, \([x]\) denotes the greatest integer less than or equal to \(x\), known as the floor function or greatest integer function.

Step 2: Analyze Discontinuity Conditions

A function is generally discontinuous where the greatest integer function has a discontinuity, which occurs at integer points. We need to explore how the values of \([x]^2\) and \([x^2]\) change as \(x\) crosses an integer.

Step 3: Evaluate at Integer Points for \([x]^2\)

When \(x\) is an integer, \(n\), \([n] = n\), so \([x]^2 = n^2\). For \(x=n+0^+\), \([x]=n\) and for \(x=n-0^+\), \([x]=n-1\). This means \([x]^2\) exhibits continuous behavior when moving through integer values.

Step 4: Evaluate at Integer Points for \([x^2]\)

For \(x\) being an integer \(n\), \(x^2 = n^2\) gives \([x^2] = n^2\). Now for small \( \epsilon > 0 \), at \(x = n + \epsilon\), \(x^2 \approx n^2 + 2n\epsilon\), thus \([x^2]\) can change if \(2n\epsilon\) crosses the next integer, which can make it discontinuous at certain points.

Step 5: Identify Continuity and Discontinuity

For non-negative integers like 0 and 1, \([x^2] = x^2\) does not cause a floor function discontinuity since its integer square itself is unaffected (\(n(n)\) does not increase). However, for integers \(n\geq2\), \([n^2 + 2n\epsilon]\) changes the integer part of the result abruptly, showing discontinuous jump in \([x^2]\).

Step 6: Determine the Correct Solution

From previous analyses, \(f(x)\) is discontinuous at all integers for integer values \(n > 1\); however, it remains continuous at \(n = 0 \) and \(n = 1\). Therefore, the correct answer is (B) all integers except 0 and 1.

Key Concepts

Floor Function

The floor function, often denoted by \[x\], is a pivotal mathematical concept. It represents the greatest integer less than or equal to any given number. This function "rounds down" a real number to the nearest integer. For example:

• For \( x = 3.7 \), \[x\] = 3.
• For \( x = -2.3 \), \[x\] = -3.

This function is defined for all real numbers and is used in a variety of mathematical contexts. The output is always an integer that is either equal to the input if the input is itself an integer or less. As we analyze the function \( f(x) = [x]^2 - [x^2] \), the floor function appears both in \( [x] \) and \( [x^2] \), offering a glimpse into its behavior when performing operations on integers and real numbers alike. The floor function is key in determining discontinuities within functions.

Greatest Integer Function

Closely related to the floor function, the greatest integer function also returns the greatest integer less than or equal to a given number. It is, essentially, another way to refer to the floor function. Notably, the greatest integer function steps down whenever the argument crosses an integer boundary. For instance:

• \( [3] = 3 \) and \( [3-0^+] = 2 \).

This jumping characteristic is essential when discussing discontinuities in function analysis. In the example of \( f(x) = [x]^2 - [x^2] \), understanding how both \( [x]^2 \) and \( [x^2] \) treat incremental changes around integer values is pivotal to sorting out continuity and potential jumps. While \( [x]^2 \) shows continuity over integers, \( [x^2] \) may exhibit sudden changes as \( x^2 \) nudges past integer squares, highlighting the greatest integer function's role in defining those deviations.

Discontinuity in Functions

Discontinuity in functions is where a function jumps, breaks, or is not smooth. With respect to the floor or greatest integer function, discontinuities are typical at integer points. Since these functions can shift values dramatically when crossing an integer, the resulting function may not smoothly transition at these points. Specifically, for \( f(x) = [x]^2 - [x^2] \):

• At integer values, the behavior of \( [x]^2 \) remains stable, while \( [x^2] \) may not, due to changes in \( x^2 \) itself.
• As \( x = n + \epsilon \), even a minute \( \epsilon \) can cause \( [x^2] \) to vary if \( 2n\epsilon \), an increment over \( n^2 \), bypasses a whole number.

The core takeaway is that this function becomes discontinuous at most integers, but not all. For figures not disrupting the integer square itself, such as \( n = 0 \) or \( n = 1 \), the floor action of \( [x^2] \) and the greatest integer evaluation remain constant, establishing the points of continuity amidst generalized discontinuous behavior.