Problem 57 - 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: Understanding the Function Components

The function given is \( f(x) = [x]^2 - [x^2] \). Here, \([x]\) represents the greatest integer less than or equal to \(x\), also known as the floor function. \([x^2]\) is the greatest integer less than or equal to \(x^2\). Our goal is to analyze the behavior of this function at integer points.

Step 2: Checking Discontinuity at Integer Points

To find where the function is discontinuous, we analyze its behavior around integers. Consider integer \(n\), and observe the function values as \(x\) approaches \(n\) from both sides. For \(x = n\), \([x] = n\), and \([x^2] \) becomes \(n^2\). However, approaching \(n\), for \(x = n^+\), \([x] = n\), but \([x^2]\) may not equal \(n^2\).

Key Concepts

Greatest Integer Function

The greatest integer function, often symbolized as \([x]\), is a mathematical function that outputs the largest integer less than or equal to a given number \(x\). It is also known as the "floor function." For instance, the input \(x=3.7\) would result in the greatest integer function outputting \([3.7] = 3\). Similarly, if \(x=-1.2\), then \([-1.2] = -2\). This concept is a cornerstone in understanding discrete mathematics because it deals with integer outcomes based on non-integer inputs.

• For any integer input \(x\), the output of \([x]\) is simply \(x\).
• If \(x\) is not an integer, the output is the highest integer less than \(x\).

In the context of the function \(f(x) = [x]^2 - [x^2]\), understanding how \([x]\) works is crucial. The expression \([x]^2\) involves taking the floor of \(x\) first and then squaring it, while \([x^2]\) involves squaring \(x\) first and then applying the floor function. These operations can yield quite different results based on the input \(x\).

Floor Function Behavior

The behavior of the floor function can be intriguing especially around integer boundaries. When analyzing functions like \(f(x) = [x]^2 - [x^2]\), grasping floor function behavior is key. When \(x\) is not exactly an integer, \([x]\) varies as \(x\) approaches any integer from either side.

• As \(x\) approaches an integer \(n\) from the left, say from \(n^-\), \([x] = n-1\).
• Conversely, approaching from the right, denoted \(n^+\), \([x] = n\).
• The floor of a squared value \([x^2]\) can differ particularly near integers due to the square's effect.

Because of these behaviors, functions involving floor operations like \(f(x)\) can have discontinuities particularly at integer values where these abrupt changes happen. Practically, this means the function does not smoothly pass through each consecutive integer.

Analyzing Function Behavior at Integers

To determine where \(f(x) = [x]^2 - [x^2]\) is discontinuous, one must specifically look at integer points. At these points, the function's value may shift substantially due to the properties of the floor function. Evaluating \(f(x)\) as \(x\) approaches an integer \(n\) involves considering values slightly less than \(n\) as well as slightly more than \(n\).

• At \(x = n^-\) — just before an integer \(n\), \([x] = n - 1\) and \([x^2]\) might reflect the previous whole number squared.
• At \(x = n\) or \(x = n^+\) — the function suddenly uses \([x] = n\) and \([x^2]\) might now take on \(n^2\), causing contrast.

Identifying when this function is discontinuous depends on recognizing these shifts. Specifically, it is shown in the exercise that discontinuity happens at intall integer values except at particular points where these evaluations become equal or close enough over the surrounding values.

To conclude, zero and one exhibit special behavior where such discontinuities can be absent, which helps clarify the original multiple-choice query.