Problem 125 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f: R \rightarrow R\) is a function defined by \(f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([x]\) denotes the greatest integer function, 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

The function is discontinuous only at non-zero integral values of \(x\). (Option C)

Step 1: Understanding the function

The function \( f(x) = [x] \cos \left(\frac{2x-1}{2}\right) \pi \) involves both the greatest integer function \([x]\) and a trigonometric component. The greatest integer function \([x]\) (also known as the floor function) returns the largest integer less than or equal to \(x\). The trigonometric part \(\cos \left(\frac{2x-1}{2}\right) \pi\) is dependent on the value of \(x\).

Step 2: Determine points of discontinuity for the greatest integer function

The function \([x]\) is discontinuous at integer values of \(x\), i.e., when \(x\) is an integer, because there is a jump from \([x]-1\) to \([x]\). Thus, we expect \(f(x)\) to be potentially discontinuous at integer values of \(x\).

Step 3: Analyzing the continuity at non-zero integral values of \(x\)

At non-zero integer values, say \(x = n\) where \(n eq 0\) is an integer, the function \(f\) switches from \([n-1] = n-1\) to \([n] = n\). Due to this shift and the behavior of the cosine function, the function could exhibit a discontinuity at these points.

Step 4: Checking continuity at zero

The function remains continuous at \(x=0\) because for \(x\) approaching zero from either side, the values of \(\cos \left(\frac{2x-1}{2}\right) \pi\) will adjust such that the continuity remains preserved. In this case, \([x]\) does not abruptly change as it does at non-zero integers, thereby suggesting continuity at \(x=0\).

Step 5: Conclusion on continuity

The function is discontinuous wherever the greatest integer function experiences a jump, other than at zero, where there is no such discontinuity. Therefore, \(f\) is discontinuous at all non-zero integer values of \(x\).

Key Concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is one of the simplest yet intriguing mathematical tools. It takes any real number and maps it to the largest integer that is less than or equal to it. For example, applying this function to 3.7 yields the result 3, and applying it to -2.3 gives -3. This function is usually denoted as \([x]\), where the square brackets indicate the greatest integer.This function is characterized by its "step-like" graph. At every integer point, there is a sudden jump from \([n-1]\) to \[n\]. These jumps indicate points of discontinuity. For functions where the greatest integer function features as a component, such jumps play a crucial role in determining the overall continuity of the function. In essence, whenever \([x]\) is used within a function, you should expect potential discontinuities at every integer value of \(x\).

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, often used to describe wave-like patterns and oscillations. The function \(\cos\left(\frac{2x-1}{2}\right)\pi\) is a trigonometric function that relies on the transformation of the input variable \(x\). The presence of the factor \(\pi\) further indicates that this is based on the standard periodic properties of cosine.The cosine function \(\cos(\theta)\) oscillates between -1 and 1 with a period of \(2\pi\). By modifying the argument of cosine, like \((\frac{2x-1}{2})\pi\), we effectively alter its frequency and phase shift. This trigonometric component plays an essential role in the function \(f(x)\) by modulating the output of the greatest integer function over its domain, influencing continuity and discontinuity patterns.Additionally, trigonometric functions are known for their continuous nature across all real numbers, which means they do not inherently introduce new discontinuities unless they are interacting with other functions that have such points.

Discontinuity Analysis

Analyzing the discontinuity of a function requires understanding how its components behave at various points. In the function \(f(x) = [x] \cos\left(\frac{2x-1}{2}\right)\pi\), the potential points of discontinuity stem from the greatest integer function's step-like jumps at integer values of \(x\).For this specific function, every non-zero integer \(x\) represents a point where \([x]\) encounters a jump, potentially causing \(f(x)\) to be discontinuous. However, it is crucial to verify each point individually to confirm this behavior. At zero, the function \(f\) surprisingly remains continuous. This happens because at \(x = 0\), the contribution from the greatest integer function does not change abruptly, maintaining continuity.To properly diagnose these discontinuities:- Check integer values for jumps in \([x]\).- Consider the interplay with the trigonometric component to see if it negates any expected discontinuities.- Test continuity at specific points using limits from either side to ensure jumps are present.By thoroughly scrutinizing these elements, we can conclude that the function \(f\) is discontinuous at all non-zero integer values of \(x\), while remaining continuous at \(x = 0\). This discontinuity analysis is vital for understanding the nature of composite functions involving non-linear components.