Problem 22 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Statement 1: A function \(f: R \rightarrow R\) is continuous at \(x_{0}\) if and only if \(\lim _{x \rightarrow x_{0}} f(x)\) exists and \(\lim _{x \rightarrow x_{0}} f(x)=f\left(x_{0}\right)\) Statement 2: A function \(f: R \rightarrow R\) is discontinuous at \(x_{0}\) i and only if, \(\lim _{x \rightarrow x_{0}} f(x)\) exists and \(\lim _{x \rightarrow x_{0}} f(x) \neq f\left(x_{0}\right)\). (a) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement \(1 .\) (b) Statement 1 is false, Statement 2 is true. (c) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1 . (d) Statement 1 is true, Statement 2 is false.

Step-by-Step Solution

Answer

(a) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

Step 1: Analyze Statement 1

Statement 1 defines continuity at a point. A function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous at \( x_0 \) if and only if the limit \( \lim_{x \rightarrow x_0} f(x) \) exists and equals \( f(x_0) \). This is the definition of continuity at a point, which is true.

Step 2: Analyze Statement 2

Statement 2 describes a condition for discontinuity. A function is discontinuous at \( x_0 \) if the limit exists but is not equal to \( f(x_0) \). However, discontinuity can also occur if the limit does not exist. Thus, this statement is true only when the limit exists but is not comprehensive for all discontinuity cases.

Step 3: Determine Correct Explanation

Statement 2 provides a scenario where the limit exists but is not equal to the function's value, indicating discontinuity. Although relevant, it indirectly illustrates what happens when Statement 1 (continuity) is not satisfied, but it does not directly explain why Statement 1 is true.

Step 4: Match with Options

Statement 1 is true, and Statement 2 is not fully comprehensive but is true for specific cases of discontinuity. Statement 2 is not a direct explanation of Statement 1's correctness. Thus, option (a) is the most appropriate choice.

Key Concepts

Limits in Calculus

Limits in calculus are fundamental to understanding how functions behave near certain points, especially when discussing continuity and discontinuity. The limit of a function \( f(x) \) as \( x \) approaches a point \( x_0 \) is what the function values are getting closer to as the input values approach \( x_0 \). In mathematical notation, this is expressed as \( \lim_{x \rightarrow x_0} f(x) \).

The concept of a limit helps in determining whether a function is continuous at a point. If the limit exists and equals the function's value at that point, we can say the function is continuous there. Conversely, if the limit doesn't exist or doesn't equal the function's value, the function could be discontinuous. Limits are utilized to understand the behavior of functions, ensuring there is a predictable pattern as inputs get closer to a particular point.

Often, limits can be determined even if a function is not explicitly defined at a particular point. This is especially useful in cases of indeterminate forms or when functions exhibit non-standard behavior at certain points. Knowing how to compute and understand limits is a cornerstone for deeper calculus concepts.

Discontinuity

Discontinuity in functions occurs when a function is not continuous at a given point. This means that at some point \( x_0 \), the limit of the function does not match the actual value of the function, or the limit may not exist at all.

There are several types of discontinuity:

• Removable Discontinuity: This occurs if the limit exists but does not equal the function’s value. Often, re-defining the function at the point can 'remove' the discontinuity.
• Jump Discontinuity: Here, the limit does not exist because the function approaches different values from different directions as \( x \) approaches \( x_0 \).
• Infinite Discontinuity: When the function approaches infinity as \( x \) approaches \( x_0 \), the function becomes discontinuous.

Understanding discontinuity is crucial because it highlights points where a function behaves unexpectedly. This knowledge is vital in real-world applications where precise modeling is important, such as engineering and physics. Identifying types of discontinuity helps both in theoretical study and practical applications.

Real-Valued Functions

Real-valued functions are functions that take real numbers as input and provide real numbers as output. These functions are central to calculus and real analysis and commonly represented as \( f: \mathbb{R} \rightarrow \mathbb{R} \).

Real-valued functions have a domain that encompasses all possible real input values and a range consisting of all real number outputs the function can produce. These functions are versatile and can express a wide array of both simple and complex relationships between numerical data.

In calculus, real-valued functions are analyzed to determine key properties like limits, continuity, derivatives, and integrals. These characteristics allow mathematicians and scientists to understand dynamic systems, model real-world scenarios, and solve complex problems. The behavior of real-valued functions, such as growth, decay, and oscillation, is essential for describing natural phenomena and enabling technological advancements.

Overall, real-valued functions form the backbone of many mathematical theories and applications, making them indispensable for students and practitioners alike.