Problem 57 - A Complete Resource Book in Mathematics for JEE Main

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 Greatest Integer Function

The greatest integer function, also known as the floor function, is defined as \([x]\) which returns the largest integer less than or equal to \(x\). For example, \([2.9] = 2\) and \([-3.4]=-4\). This function is known to be discontinuous at any integer \(n\) since there is a jump in values, from \( [n-\epsilon] = n-1 \) to \( [n] = n \).

Step 2: Analyze the Function Components

The given function is \( f(x) = [x]^2 - [x^2] \). Here \([x]^2\) means taking the greatest integer less than or equal to \(x\) and then squaring it, while \([x^2]\) takes \(x^2\) and returns the greatest integer less than or equal to \(x^2\). Both \([x]^2\) and \([x^2]\) are floor functions, which suggest possible discontinuities around integers.

Step 3: Check Behavior Near Integer Points

Let's determine the function's value at critical points. When \(x\) approaches an integer \(n\) from the left (i.e. \( x = n^-\)), \([x]\) is \(n-1\) and thus \([x]^2 = (n-1)^2\). When \(x\) approaches from the right (i.e. \( x = n^+\)), \([x]\) is \(n\) and \([x]^2 = n^2\). Let's see how \([x^2]\) behaves: if \(n - 1 < x < n\), then \(x^2 = n^2 - \epsilon\), yielding \([x^2] = n^2-1\). If \(n \leq x < n + 1\), then \([x^2] = n^2\). This discrepancy indicates that \(f(x)\) is discontinuous at \(n\).

Step 4: Special Cases of 0 and 1

Check to see if any exceptions exist, particularly at 0 and 1, based on the exercise options. At \(x=0\), \([x] = 0\), and \([0^2] = 0\), leading to \(f(0) = 0^2 - 0 = 0\). Approaching 0 from both directions leads to \(f(x) = 0\) continuously. At \(x=1\), \([x] = 1\), \([x^2] = [1]\) is \(1\), yielding \(f(1) = 1^2 - 1 = 0\), with no discontinuity in either direction. Therefore, \(f(x)\) is continuous at 0 and 1.

Step 5: Conclusion with Given Options

Since we've determined the function is discontinuous at the majority of integers, except where it behaves continuously at 0 and 1, the correct answer is (B) all integers except 0 and 1.

Key Concepts

greatest integer function

The greatest integer function, often called the floor function, is represented as \[ x \.\]. This function gives you the largest integer less than or equal to a given number. It's quite straightforward: if you feed it a decimal or fraction, it will "round down" to the nearest whole number. For example:

• When you input 2.9, it will output 2.
• For -3.4, the result will be -4, as it rounds down to the next integer.

This function is key in mathematics for statistical data rounding, among other applications. Every integer point creates a "jump" or shift in the result, which is essential in understanding discontinuous functions.

floor function

The floor function, symbolized as \[ \lfloor x \rfloor \,\], holds the same meaning as the greatest integer function. Its purpose is to convert a real number into the greatest integer that is less than or equal to that number. This conversion leads to unique behavior at integer points:

• For example, \( \lfloor 2.7 \rfloor \) computes to 2.
• Similarly, \( \lfloor -1.2 \rfloor = -2 \).

The floor function is often paired with graphical analysis to depict discontinuities in a function's behavior, as seen in many real-world calculations.

integer discontinuities

Integer discontinuities arise when a function displays a noticeable jump or inconsistency at integer values. The floor function or greatest integer function serves as a prime example of a function with this trait. This kind of function doesn't change gradually at integer points. Consider a function with sudden shifts at integers as values "jump" from one to the next:

• The value directly before an integer, such as 1.99, results in \( [1.99] = 1 \).
• Once it crosses the integer value, such as 2.01, it immediately goes up to \( [2.01] = 2 \).

These jumps make the function discontinuous at each integer, which is vital in understanding functions that aren't smooth or aren't completely predictable at every point.

function behavior analysis

Function behavior analysis involves examining how a function behaves around specific critical points, often focused on its continuity and discontinuity. In our example with the greatest integer function, the key is closely inspecting the transitions at and around integer values. Certain functions can switch character from smooth (continuous) to jumpy (discontinuous) depending on the position. Here's why behavior analysis matters:

• Understanding where a function has spikes or gaps helps mathematicians and engineers predict and manipulate systems effectively.
• By focusing on how functions like \( f(x) = [x]^2 - [x^2] \) behave around integers, we know when and where to expect interruptions.

This analysis aids in solving optimization problems or understanding scenarios in physics or coding, where such discontinuities may affect outcomes significantly.