Problem 9 - 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

The function is discontinuous at all integers except 0 (Option C).

Step 1: Understand the Function Components

The given function is \( f(x) = [x]^2 - [x^2] \). Here, \([x]\) is the greatest integer function, which returns the largest integer less than or equal to \(x\). We need to determine where \(f(x)\) is discontinuous.

Step 2: Analyze Discontinuity at Integers

A function \( f(x) \) using the greatest integer function is usually discontinuous at integers. We consider the behavior of \([x]^2\) and \([x^2]\) separately for integers.

Step 3: Evaluate at Specific Points

Let’s analyze the function at integer points, particularly checking at 0 and 1 separately, as suggested by options: - When \( x \) is an integer, \([x] = x\), and \([x]^2 = x^2\). - If \(x = n\) where \(n\) an integer, \([n^2] = n^2\). Thus, \( f(n) = n^2 - n^2 = 0\).- For integers other than 0, expressions may differ when approaching from positive or negative sides.

Step 4: Check Behavior around Nonzero Integers

Consider one-sided limits around nonzero integers: - For \( x = 1^+ \), \([x] = 1\) and \([x^2] = 1\). As \(x \rightarrow 1^- \), \([x] = 0\) and \([x^2] = 0\). - There is no discontinuity at 0 because approaching from both sides does not result in differing expressions as integers around 0 are negative (i.e., \([-1]\) who squares still result in 0).

Step 5: Determine Discontinuity

The function is discontinuous at all integers except 0, confirming that it behaves continuously around nonzero integer inputs when approached from both sides near any integer except where |\(x > 1\) or \(x < 1\).

Key Concepts

Greatest Integer Function

The greatest integer function, denoted as \([x]\), plays a crucial role in understanding the behavior of functions that involve rounding numbers down to the nearest integer. This function returns the largest integer less or equal to a given number \(x\). For example:

• \([2.7] = 2\)
• \([-1.3] = -2\)
• \([5] = 5\)

This step function creates discontinuities, notably at integer points, because the greatest integer function jumps from one integer to the next. Thus, you can expect distinct changes in value as you cross integer boundaries, directly impacting the overall continuity of any function using \([x]\). In our specific exercise, the function \(f(x) = [x]^2 - [x^2]\) depends heavily on this behavior to determine points of discontinuity. By recognizing where the jumps occur, we can see how they contribute to how \(f(x)\) behaves near integer values.

Integer Points Analysis

Integer points play a significant role when analyzing functions that contain the greatest integer function. At each integer point, the behavior of the function can dramatically shift due to the nature of the step function.To analyze such functions, consider the behavior of the individual components at integer points:

• For \([x]^2\), if \(x\) is an integer, the function remains consistent, as it evaluates to \(x^2\).
• For \([x^2]\), when \(x\) is an integer, it directly evaluates to \(x^2\), providing a consistent output.
• Therefore, at any integer \(n\), \(f(n) = [n]^2 - [n^2] = n^2 - n^2 = 0\).

Despite this consistency at integer values themselves, scrutiny of the surrounding behavior is key. The function's difference in behavior right before and after integer crossings impacts its overall discontinuity. This kind of analysis reveals where changes in the function’s values result in removable or removable discontinuities, thus providing deeper insight into where continuity breaks might occur in a function like \(f(x)\).

One-Sided Limits

One-sided limits assist in revealing where a function like \(f(x) = [x]^2 - [x^2]\) may be discontinuous around integer points. These limits involve approaching an integer from the left or right, offering insight into the precise behavior of the function in these regions.Consider integer values such as 0 and 1:

• For \(x = 1^+\) (approaching from the right), \([x] = 1\) and \([x^2] = 1\), leading to no change.
• As \(x ightarrow 1^-\) (approaching from the left), \([x] = 0\) and \([x^2] = 0\), meaning possible differences manifest between these one-sided values.
• The consistency at \(x=0\) arises as negative values near zero (such as -0.1 becoming -1) maintain consistent expression values, unlike positive integer transitions.

Understanding one-sided limits allows you to capture subtle distinctions in a function’s behavior at boundary points. Detecting discontinuity becomes feasible, distinguishing instances where the function stays consistent across boundaries or where it exhibits abrupt changes. These insights form the basis for evaluating continuity in functions with the greatest integer components.