Problem 105 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(f\) be a function defined on an interval \(\mathrm{I}\). If \(f\) be discontinuous at a point \(p \in I\), then we say that (i) \(f\) has a removable discontinuity at \(p\) if \(\lim _{x \rightarrow p} f(x)\) exists but is not equal to \(f(p)\). (ii) \(f\) has a discontinuity of the first kind at \(p\) if \(\lim _{x \rightarrow p-0} f(x)\) and \(\lim _{x \rightarrow p+0} f(x)\) exist but are unequal. (iii) \(f\) has a discontinuity of the second kind at \(p\) if neither of \(\lim _{x \rightarrow p-0} f(x)\) and \(\lim _{x \rightarrow p+0} f(x)\) exists. Let \(f(x)=\frac{1}{[\cos x]}\), where \([\cdot]\) denotes the greatest integer function. Then, the function \(f(x)\) has at \(x=\frac{\pi}{2}\) (A) removable discontinuity (B) discontinuity of first kind from left (C) discontinuity of second kind from left (D) None of these

Step-by-Step Solution

Answer

Option C; discontinuity of second kind from left.

Step 1: Understanding the Function

The function given is \( f(x) = \frac{1}{[\cos x]} \), where \([\cdot]\) is the greatest integer function. This means the denominator is the greatest integer less than or equal to \( \cos x \). We need to assess the behavior of \( f(x) \) at \( x = \frac{\pi}{2} \).

Step 2: Evaluating \( \cos \frac{\pi}{2} \)

Calculate \( \cos \frac{\pi}{2} \). We know that \( \cos \frac{\pi}{2} = 0 \).

Step 3: Applying the Greatest Integer Function

Since \( \cos \frac{\pi}{2} = 0 \), \( [\cos \frac{\pi}{2}] = 0 \) as the greatest integer less than or equal to zero is zero. Thus, \( f(x) = \frac{1}{0} \) at \( x = \frac{\pi}{2} \), which is undefined.

Step 4: Approaching from the Left

Consider \( x \to \frac{\pi}{2}^- \). Here, \( \cos x \to 0^+ \), meaning the values of \( \cos x \) approaches zero from the right. For values just less than \( \frac{\pi}{2} \), \([\cos x] = 0\), hence \( f(x) \) is undefined approaching from the left.

Step 5: Approaching from the Right

Consider \( x \to \frac{\pi}{2}^+ \). Here, \( \cos x \to 0^- \), meaning the values of \( \cos x \) approach zero from the left. For values just greater than \( \frac{\pi}{2} \), \([\cos x] = -1\). Thus, \( f(x) = -1 \) approaching from the right.

Step 6: Analyzing the Discontinuity

We see that from the left, \( f(x) \) is undefined and from the right, \( f(x) = -1 \). Since \( f(x) \) is undefined from one side (left), neither of the one-sided limits exists. Thus, \( f \) has a discontinuity of the second kind at \( x = \frac{\pi}{2} \).

Key Concepts

Removable Discontinuity

A removable discontinuity is a unique type of discontinuity that occurs in a function when the limit of the function as it approaches a specific point exists. However, this limit does not equal the value of the function at that particular point. In essence, it is like the function almost connects to a specific point, but there is a hole or missing spot right at that connection.

To make this concept more tangible, consider a function that behaves nicely as it approaches a specific value, yet, at that precise value, something unexpected disrupts the continuity. Luckily, with some adjustments, such missing links can often be "removed," hence the term removable discontinuity.

The adjustment usually involves redefining the value of the function at the discontinuous point to be the same as the surrounding limit. Understanding this concept provides a valuable insight into graphing functions and diagnosing issues within them that can be mended with minor modifications.

Discontinuity of the First Kind

Discontinuity of the first kind occurs when the limits of a function from both sides of a particular point exist, but these one-sided limits are not equal to each other. There is a sudden jump or step at the discontinuous point, making it appear as though the graph is broken and needs to be reconnected.

When visualizing this kind of discontinuity, envision a function graph where the path makes an abrupt leap at the discontinuous point, moving immediately to another level. This situation reveals a function behaving differently on either side of a point, calling this a jump discontinuity as well.

In practice, identifying this discontinuity helps highlight where a function might switch behaviors or deviate abruptly, allowing us to dissect and understand the underlying behavior of various mathematical models or real-world data representations.

Discontinuity of the Second Kind

Discontinuity of the second kind is more challenging to handle as neither one-sided limit exists at the discontinuity point of the function. In such cases, the behavior of the function becomes erratic or unpredictable as it approaches the point from either direction.

This type of discontinuity signifies an area in a graph where the function does something wildly different as it climbs or descends toward a value. Unlike the first kind, here no limit can be defined from either the left or right, making the function completely undefined at the point of interest.

Understanding this discontinuity is crucial when examining functions that exhibit severe irregular behavior, where usual methods of redefinition or smoothing out do not apply. These discontinuities often appear in complex functions involving oscillations or infinite behaviors, requiring careful analysis to interpret or use these functions effectively.