Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f: R \rightarrow R\) is a fucntion defined by \(f(x)=[x] \cos\) \(\left(\frac{2 x-1}{2}\right) \pi\), where \([x]\) denotes the greatest integer fucntion, then \(f\) is (A) continuous for every real \(x\) (B) discontinuous only at \(x=0\) (C) discontinuous only at non-zero integral values of \(x\) (D) continuous only at \(x=0\)

Step-by-Step Solution

Answer

(C) discontinuous only at non-zero integral values of \( x \)

Step 1: Understanding the Function

The function given is \( f(x) = [x] \cos \left( \frac{2x-1}{2} \right) \pi \), where \([x]\) represents the greatest integer less than or equal to \(x\). The cosine function results in specific values which help in determining continuity at certain points.

Step 2: Behavior at Integer Values of x

At integer values of \( x \), say \( n \), \( f(n) = n \cos \left( \frac{2n-1}{2} \right) \pi \). The cosine part can result in 0 for half-integer values \( (2n-1)/2 \) because the cosine function \( \cos((2n-1)\pi/2) \) results in zero for odd \( n \). Thus, \( f(n) = 0 \). However, a just-left or just-right point of \( n \) like \( n-0.1 \) or \( n+0.1 \) would evaluate to different values, creating discontinuity.

Step 3: Behavior at Non-Integer Values

For non-integer values of \( x \), \([x]\) is not changing rapidly as at integer values. The term \( [x]\) remains constant between two consecutive integers and cosine varies smoothly unless the argument is a half-integer. This indicates that non-integers naturally do not pose issues of continuity despite the complexity of the cosine term.

Step 4: Checking Discontinuity at Zero

At \( x = 0 \), \([0] \) is zero, thus \( f(0) = 0 \). Closely approaching zero from either side would still use the floor value of an integer portion, hence no abrupt change occurs, suggesting continuity at \( x = 0 \).

Step 5: Determine Where Discontinuities Occur

Based on these evaluations, the function \( f(x) = [x] \cos \left( \frac{2x-1}{2} \right) \pi \) is discontinuous at every integer except zero, since near every integer it jumps due to change in \([x]\) while staying continuous at non-integers.

Key Concepts

Greatest Integer Function

The Greatest Integer Function, often represented by \([x]\), is a step function that floors a real number to the nearest integer less than or equal to it. For example, if the input is 3.7, then \([3.7] = 3\). This property causes \([x]\) to create an inherent discontinuity at every integer point on the real line because the function jumps from one integer to the next as the input value crosses an integer.

• Key characteristic: The output jumps as the input crosses an integer value.
• Example: \([2.9] = 2\) but \([3.1] = 3\), indicating a step change.
• This jump or change is what introduces discontinuity in the function, particularly at non-zero integer points when paired with other mathematical functions, such as trigonometric expressions.

Understanding how this function behaves is crucial when analyzing continuity in problems where it's involved, like the given exercise.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics for relating angles to side lengths in right-angled triangles, and in this context, we're particularly interested in the cosine function. Cosine, written as \( \cos(x)\), is periodic with a period of \((2\pi)\), meaning it repeats its pattern every \((2\pi)\) units.

• Key values: \( \cos(0) = 1, \cos(\pi) = -1, \cos(\frac{\pi}{2}) = 0, \cos(\frac{3\pi}{2}) = 0 \).
• It provides continuity and smoothness, except when combined with certain expressions that might introduce discontinuity.
• Especially important is how it behaves when its argument shifts slightly, and noting that it can be zero when the argument is an odd multiple of \((\pi/2)\).

In the function from the exercise, the argument of the cosine function \((2x-1)/2\) introduces half-integer multiples, which cause the function to touch zero at specific intervals, enhancing discontinuity when compounded with the greatest integer function.

Real Valued Function Analysis

Real-valued functions take real numbers as inputs and provide real numbers as outputs. When we're analyzing such functions, especially for continuity and discontinuity, understanding how both the primary function and any piecewise components (like those seen with the greatest integer function) interact is vital.

• Continuity: A function is continuous at a point if small changes in the input result in small changes in the output.
• Discontinuity arises when this smooth transition breaks, often at sudden jumps or asymptotes.
• Real-valued functions such as \( f(x) = [x] \cos((2x-1)\pi/2) \) exhibit discontinuity chiefly at integer values of \( x \), due to the combination of discontinuous step functions and trigonometric fluctuations.

In the provided exercise, critical analysis of these functions and their behavior at specific points helped identify where continuity holds and where it fails, confirming that discontinuity occurs at non-zero integers while zero remains continuous.