Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The function \(f(x)=[x] \cos \left(\frac{2 x-1}{2}\right) \pi\), where \([.]\) denotes the greatest integer function, is discontinuous at (A) all \(x\) (B) all integer points (C) no \(x\) (D) \(x\) which is not an integer

Step-by-Step Solution

Answer

The function is discontinuous at all integer points, choice (B).

Step 1: Understanding the Greatest Integer Function

The greatest integer function, denoted as \([x]\), represents the largest integer less than or equal to \(x\). This means \([x]=n\) if \(n \leq x < n+1\), where \(n\) is an integer.

Step 2: Analyzing Discontinuity in Terms of Integers

To determine if \(f(x)=[x]\cos\left(\frac{2x-1}{2}\right)\pi\) is discontinuous at integer points, consider that \([x]\) changes its value at every integer. Therefore, \(f(x)\) could be potentially discontinuous at integers due to the change in \([x]\).

Step 3: Evaluating the Cosine Component

The term \( \cos\left(\frac{2x-1}{2}\right)\pi \) remains continuous as it's a trigonometric function. However, discontinuity in \(f(x)\) can occur due to the presence of the greatest integer function, especially at points where \(x\) is an integer.

Step 4: Testing for Discontinuity at Integer Points

At integer points, \(x=n\), \([x]=n\). As we approach from the left, \(x=n^-\), \([n^-]=n-1\) and \(x=n^+\), \([n^+]=n\), indicating that there's a jump in the function value, showing discontinuity.

Step 5: Concluding Discontinuity Condition

The function \(f(x)\) is discontinuous where \([x]\) changes, which occurs at all integer points. The correct choice is (B) all integer points, as this is where the discontinuity due to the greatest integer function occurs.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([x]\), is a fascinating mathematical concept. It represents the largest integer that is less than or equal to a given number \(x\). Essentially, it rounds down a real number to the nearest whole number lower than or equal to it.

• For example, if \(x = 2.3\), then \([x] = 2\).
• If \(x = -2.3\), then \([x] = -3\). This is because \(-3\) is the greatest integer less than or equal to \(-2.3\).

A key feature of this function is how it behaves at integer values. At any integer point \(n\), both \([n^-]\) and \([n^+]\) represent the floor of the number, resulting in a jump between these two values. This characteristic causes what is known as a 'jump discontinuity' at every integer point. Thus, the greatest integer function is inherently discontinuous at any point where \(x\) is a whole number.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, play a crucial role in mathematics. For the function \(f(x)=[x] \cos\left(\frac{2x-1}{2}\right)\pi\), the cosine component is a key player. The cosine function represents a periodic wave, continuously oscillating between 1 and -1.

• The cosine function is continuous and smooth along the entire real line.
• It maintains continuity because it does not have jumps or interruptions in its values.

In the given function, the term \(\cos\left(\frac{2x-1}{2}\right)\pi\) ensures that the oscillation remains smooth as \(x\) changes. However, its continuity is overpowered by the discontinuous nature of the greatest integer function at integer points. The presence of this function does not introduce any additional discontinuities to the function \(f(x)\), aside from those already imposed by \([x]\).

Continuity and Limits

Continuity and limits are fundamental concepts in calculus used to analyze the behavior of functions at certain points. A function is continuous at a point if it does not have any abrupt changes or jumps there, meaning you can draw its graph without lifting your pen at that point.

• A limit describes the value that a function approaches as the input approaches some value.
• For continuity at a point \(c\), the limit of \(f(x)\) as \(x\) approaches \(c\) from both directions should equal \(f(c)\).

In the case of \(f(x)= [x] \cos\left(\frac{2x-1}{2}\right)\pi\), we observe discontinuity at integer points due to \([x]\). At any integer \(n\), the function value abruptly jumps, as seen when \(x\) approaches \(n\) from the left and right. The value from the left, \([n^-]\), differs from the value from the right, \([n^+]\), indicating a break in the continuity. Therefore, limits are particularly useful in identifying and understanding such discontinuities within functions that involve the greatest integer function.