Problem 72 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f\) is a continuous function from \(R\) to \(R\) and \(f(f(a))=a\) for some \(a \in R\), then the equation \(f(x)=x\) has (A) no solution (B) exactly one solution (C) at most one solution (D) at least one solution

Step-by-Step Solution

Answer

(D) at least one solution

Step 1: Understanding the Function Condition

We are given that \( f(f(a)) = a \) for some \( a \in \mathbb{R} \). This implies that \( f \) is an involution, meaning applying \( f \) twice results in the original input. Continuity of the function \( f \) is also given.

Step 2: Assuming a Solution Exists

If there is a solution \( x \) to \( f(x) = x \), then we must have \( f(f(x)) = f(x) = x \). This satisfies the condition \( f(f(a)) = a \). Therefore, if \( x = a \), \( f(x) = x \) is satisfied.

Step 3: Existence of a Fixed Point

Since \( f \) is continuous, and we see that applying \( f \) twice returns the original input, there must be a point which maps to itself, that is, \( f(x) = x \). This implies that due to the continuity and the involution property, there must be at least one point where this condition is satisfied.

Step 4: Conclusion on Solutions

As shown, due to the properties of continuity and the fact that \( f(f(a)) = a \), there must be at least one fixed point where \( f(x) = x \). Therefore, the assertion that the equation has at least one solution is valid.

Key Concepts

Continuous Functions

Continuous functions are a fundamental concept in real analysis. They are functions that, informally speaking, have no sudden jumps. This means that small changes in the input result in small changes in the output. More formally, a function \( f: \mathbb{R} \to \mathbb{R} \) is continuous at a point \( c \) if \( \lim_{x \to c} f(x) = f(c) \). If this is true for every point in the domain, then \( f \) is said to be continuous everywhere.
Continuous functions have some very useful properties in mathematical analysis:

• They can be neatly graphed without lifting your pencil from the paper.
• They satisfy the intermediate value theorem, meaning if a continuous function takes values \( f(a) \) and \( f(b) \), it must take any value between \( f(a) \) and \( f(b) \) at some point between \( a \) and \( b \).
• They appear in many real-world scenarios like physics and engineering where smooth transitions are necessary.

Understanding continuous functions helps grasp why some equations always have at least one solution, such as those in fixed point theorems.

Involution

An involution is a very interesting concept in real analysis, especially when dealing with functions. Let's break it down: an involution is a function \( f \) such that \( f(f(x)) = x \) for all \( x \) in its domain.
In other words, if you apply the function twice, you get back to your starting point. Here's an example to illustrate:

• Consider the function \( f(x) = -x \). Here, \( f(f(x)) = f(-x) = -(-x) = x \). Thus, \( f \) is an involution.

Involution functions often reveal symmetrical properties within a mathematical system. They can serve to invert processes or reverse operations. Important involutions include reflections and rotations in geometry and even certain permutations in algebra.
This concept plays a crucial role when examining situations that inherently involve reversibility, such as involutions ensuring at least one fixed point in continuous functions as the exercise suggests.

Real Analysis

Real analysis is a branch of mathematical analysis dealing with real numbers and real-valued functions. It’s one of the core subjects in advanced mathematics, providing the theoretical underpinnings for calculus and other significant fields.
Here are some critical areas it covers:

• Sequences and series: Real analysis rigorously defines convergence and divergence of sequences and series.
• Limits and continuity: It offers a precise understanding of limits, continuity, and differentiability.
• Integration: Real analysis explores Lebesgue and Riemann integration to generalize standard notions of integrating functions.

Moreover, real analysis considers the structure of the real number line and includes sophisticated theorems like the Bolzano-Weierstrass and the Cantor intersection theorem, among others. Mastery of real analysis is key for students wanting to delve deeper into higher mathematics, as many concepts in real analysis are foundational for other fields such as statistics, quantum mechanics, and more. By understanding these core pieces, students can better comprehend how and why certain theorems, like fixed point theorems, hold when conditions like continuity are met.