Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=\left\\{\begin{array}{l}x, \text { when } x \text { is rational } \\\ 1-x, \text { when } x \text { is irrational }\end{array}\right.\), then (A) \(f(x)\) is continuous for all real \(x\) (B) \(f(x)\) is discontinuous for all real \(x\) (C) \(f(x)\) is continuous only at \(x=1 / 2\) (D) \(f(x)\) is discontinuous only at \(x=1 / 2\).

Step-by-Step Solution

Answer

(C) \(f(x)\) is continuous only at \(x = 1/2\).

Step 1: Understand the Function

The function is defined piecewise. For rational numbers, the function is simply \( f(x) = x \). For irrational numbers, the function is \( f(x) = 1-x \). This means that the function behaves differently depending on whether \( x \) is rational or irrational.

Step 2: Concept of Continuity

For a function to be continuous at a point \( a \), three conditions must be met: \( f(a) \) is defined, \( \lim_{x \to a} f(x) \) exists, and \( \lim_{x \to a} f(x) = f(a) \). Evaluate whether these conditions are met for all points or specific points.

Step 3: Examine Continuity at x = 1/2

At \( x = 1/2 \), \( f(1/2) = 1/2 \) for both rational and irrational \( x \), since \( f(x) = x \) when rational and \( f(x) = 1-x \) when irrational both evaluate to \( 1/2 \). Both limits (from rational and irrational directions) equal \( 1/2 \), hence the function is continuous at \( x=1/2 \).

Step 4: Check Other Points

For any other \( x eq 1/2 \), if \( x \) is rational, the function is \( f(x) = x \), and if \( x \) is irrational, the function is \( f(x) = 1-x \). The limits from rational and irrational sequences do not match, hence \( f(x) \) is discontinuous at all other points.

Key Concepts

Rational Functions

Rational functions are mathematical expressions that can be represented as the quotient of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials, and \( Q(x) eq 0 \).

These functions exhibit interesting characteristics due to their ability to have vertical asymptotes, horizontal asymptotes, and other unique behaviors around their undefined points. Vertical asymptotes occur where the denominator equals zero, leading to a division by zero.

Rational functions are important in calculus as they often demonstrate limits and continuity concepts. A rational function may not be continuous at certain points due to these asymptotes. Understanding how to identify vertical asymptotes and discontinuities in rational functions is crucial when analyzing them for calculus applications.

Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be written as \( \frac{a}{b} \), where \( a \) and \( b \) are integers. These numbers have non-repeating, non-terminating decimal expansions, like \( \sqrt{2} \) and \( \pi \).

In the context of piecewise functions, irrational numbers play a significant role as they can form separate branches in the function's definition. For example, in a piecewise function that distinguishes between rational and irrational inputs, irrational inputs can lead to completely different evaluations compared to rational inputs.

Understanding irrational numbers is essential when analyzing piecewise functions, as these functions often need to meet specific conditions depending on the irrationality or rationality of the input. Delving deeper into the characteristics of irrational numbers helps provide insight into the continuity and behavior of such functions.

Limits in Calculus

Limits are a fundamental concept in calculus used to understand the behavior of functions as they approach specific points. Mathematically, the limit of a function \( f(x) \) as \( x \) approaches \( a \) is denoted as \( \lim_{x \to a} f(x) \).

Limits are crucial for determining continuity. A function is continuous at a point \( a \) if the following three conditions are met:

• \( f(a) \) is defined
• The limit of \( f(x) \) as \( x \to a \) exists
• The limit of \( f(x) \) as \( x \to a \) equals \( f(a) \)

In piecewise functions like the one in the original exercise, evaluating limits from both rational and irrational directions is necessary to fully assess the function's continuity.

Understanding limits helps analyze how functions behave near points of interest and aids in solving many advanced calculus problems, such as determining derivatives and integrals.