Question

Let \([x]\) be the greatest integer less than or equals to \(x\). Then, at which of the following point \((\mathrm{s})\) the function \(f(x)=x \cos (\pi(x+[x]))\) is discontinuous? [A] \(x=-1\) [B] \(x=0\) [C] \(x=1\) [D] \(x=2\)

Short answer

The function is discontinuous at the points \(x=-1\), \(x=1\), and \(x=2\).

Step-by-step solution

Understanding the Greatest Integer Function

The greatest integer function, denoted as \[x\], returns the greatest integer less than or equal to \(x\). This function is also known as the floor function. It causes discontinuities in any function it's a part of wherever \(x\) is an integer because the function steps down to the next integer value.

Analyzing Discontinuities in Combination with Cosine

Since \(\cos(\pi(x + [x]))\) always yields a value between -1 and 1, it will not, by itself, be the cause of a discontinuity. However, its value can be 0 when \(x+[x]\) is an odd half-integer, which is important because multiplying by zero could make \(f(x)\) continuous at points where \(x\) is an integer. Whenever \(x\) is an integer, \(f(x)\) could possibly be discontinuous because of the \[x\] function, unless the cosine term equals zero, negating the effect.

Evaluating the Discontinuity at \(x=-1\)

At \(x=-1\), the function \(f(x)\) simplifies to \(f(-1) = -1 \times \cos(\pi(-1+[-1])) = -1 \times \cos(\pi(-1-1)) = -1 \times \cos(-2\pi) = -1 \times 1 = -1\). The step in the greatest integer function comes right before \(x=-1\), which causes a discontinuity in \(f(x)\).

Evaluating the Discontinuity at \(x=0\)

At \(x=0\), the greatest integer function evaluates to \[0\] and \(f(0)\) becomes \(f(0) = 0 \times \cos(\pi(0+[0])) = 0\), which is continuous since the function \(f(x)\) doesn't change value abruptly at \(x=0\).

Evaluating the Discontinuity at \(x=1\)

At \(x=1\), the function \(f(x)\) simplifies to \(f(1) = 1 \times \cos(\pi(1+[1])) = 1 \times \cos(\pi(1+1)) = 1 \times \cos(2\pi) = 1\). This indicates a potential discontinuity since the value of the greatest integer function changes at \(x=1\).

Evaluating the Discontinuity at \(x=2\)

At \(x=2\), the function \(f(x)\) simplifies to \(f(2) = 2 \times \cos(\pi(2+[2])) = 2 \times \cos(\pi(2+2)) = 2 \times \cos(4\pi) = 2\). This indicates a potential discontinuity since the value of the greatest integer function changes at \(x=2\).

Final Judgement of Discontinuities

We have identified possible discontinuities at \(x=-1\), \(x=1\), and \(x=2\) due to the step nature of the greatest integer function. However, the cosine function does not mitigate these discontinuities at those points. Therefore, the function is discontinuous at \(x=-1\), \(x=1\), and \(x=2\).

Key concepts

Greatest Integer Function

The greatest integer function, also known commonly as the floor function, is a step function that assigns to any real number \(x\) the largest integer less than or equal to \(x\). It is denoted as \([x]\).

This function is essential in various branches of mathematics, including algebra and number theory, and it plays a significant role in mathematical problems of competitive exams like JEE Advanced. The graph of this function consists of a series of steps, which immediately gives us a visual understanding of its discontinuous nature. These discontinuities occur at every integer value of \(x\), as the output jumps from one integer to the next without taking any value in between.

For example, when transitioning from \(x=1.9\) to \(x=2\), the greatest integer function immediately jumps from \([1.9] = 1\) to \([2] = 2\), illustrating a clear break or 'jump' in the function's graph. This distinction is crucial for students preparing for exams like JEE Advanced, where understanding the behavior of functions at every point is necessary to tackle rigorous mathematical problems.

Cosine Function

The cosine function is one of the primary trigonometric functions, depicting the ratio between the adjacent side and hypotenuse of a right-angled triangle. In the context of pure mathematics, the function \(\cos(x)\) is periodic and continuous over all real numbers, described by its wave-like graph that oscillates between -1 and 1.

The cosine curve’s shape is a key concept in JEE Advanced mathematics, where harmonic motion and alternating waveforms are common topics. Being continuous, the cosine function does not, by itself, introduce any discontinuities into the functions it is a part of. It is a smooth and unbroken function that can be helpful in mitigating discontinuities of other functions when multiplied together, such as in the provided exercise.

Understanding the periodic and continuous properties of the cosine function is instrumental for students, as the function often appears in conjunction with others in JEE Advanced and other mathematical problem-solving scenarios.

JEE Advanced Mathematics

JEE Advanced mathematics demands a deep understanding of various mathematical concepts including calculus, algebra, and trigonometry. Exam problems often combine functions in ways that challenge a student's conceptual understanding and application skills. The provided exercise intertwines the greatest integer function with the cosine function—a common practice in JEE Advanced to test a student’s grasp of function behavior and discontinuity.

In JEE Advanced mathematics, questions are designed to evaluate students’ proficiency in analysis and problem-solving. The ability to dissect a complex function into simpler parts and examine their individual behaviors, as exemplified in the step-by-step solution provided, is an essential skill. As such, students are advised to master functions like the ‘greatest integer function’ and ‘cosine function’, as well as their properties, to excel in the exam.

By recognizing where and why a function like \(f(x) = x \cos(\pi(x + [x]))\) is discontinuous, students can better approach and solve complex problems, a critical element in performing well in JEE Advanced mathematics and similar high-level math examinations.