Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let a function \(f: R \rightarrow R\) satisfy the equation \(f(x+y)=\) \(f(x)+f(y)\) for all \(x, y\). If the function \(f(x)\) is continuous at \(x=0\), then (A) \(f(x)=0\) continuous for all \(x\) (B) \(f(x)\) is continuous for all positive real \(x\) (C) \(f(x)\) is continuous for all \(x\) (D) None of these

Step-by-Step Solution

Answer

Option C: \(f(x)\) is continuous for all \(x\).

Step 1: Analyze the Given Functional Equation

We start by analyzing the functional equation: \( f(x+y) = f(x) + f(y) \) for all \( x, y \). This is the Cauchy functional equation, which is known to have linear solutions of the form \( f(x) = cx \) where \( c \) is a constant.

Step 2: Apply the Continuity Condition

The function \( f(x) \) is continuous at \( x = 0 \). For Cauchy functions, if the function is continuous at any point, it is a linear function. Thus, \( f(x) = cx \) for some real constant \( c \) and it must be continuous everywhere.

Step 3: Discuss Continuity of Linear Functions

Since \( f(x) = cx \) is a linear function, it is continuous for all real numbers \( x \). Linear functions are continuous everywhere because they can be expressed in the form \( f(x) = mx + b \), which has no abrupt changes or discontinuities.

Step 4: Conclude Based on Continuity Results

Given the functional equation and the continuity at \( x = 0 \), and since the function becomes linear, we conclude that \( f(x) \) is continuous for all \( x \). Therefore, option (C) is correct.

Key Concepts

Continuity

Continuity is a fundamental concept in calculus and analysis. It describes a property of functions where small changes in the input result in small changes in the output.
If a function is continuous at a point, there are no jumps, breaks, or holes at that point. This means you can draw the graph of the function through that point without lifting your pencil.

• For a function to be continuous at a specific point, say at point \(x = a\), the following must hold:
  • The function \(f(x)\) must be defined at \(x = a\).
  • The limit \( \lim_{x \to a} f(x) \) must exist.
  • The limit must equal the function value: \( \lim_{x \to a} f(x) = f(a) \).

When a function is continuous at a point and satisfies a functional equation like the Cauchy functional equation \( f(x+y) = f(x) + f(y) \), it implies a special type of simplicity in the function's structure, leading us toward linearity.

Linear Functions

Linear functions are among the simplest types of functions. They are characterized by the equation form \( f(x) = mx + b \), where \( m \) and \( b \) are constants.
The graph of a linear function is a straight line, which indicates that the change in \( f(x) \) is constant for a change in \( x \).

• The slope \( m \) defines the rate of change of the function.
• The y-intercept \( b \) indicates the point at which the graph of the function crosses the y-axis.

Linear functions are continuous everywhere because they do not have abrupt changes. They are smooth and predictable, making them an important tool in modeling a variety of real-world situations. This continuity and simplicity align well with the solutions to the Cauchy functional equation when continuity is guaranteed.

Real Analysis

Real analysis provides the rigorous underpinnings of calculus by exploring the properties of real numbers and real-valued functions. It introduces precise definitions and proofs to concepts that might seem intuitively clear.

• Core topics include sequences, limits, continuity, differentiation, and integration.
• Real analysis considers both the fine-grained details and the overall behavior of functions in terms of real numbers.

When dealing with the Cauchy functional equation, real analysis helps us understand why continuous solutions are linear. By ensuring rigorous examination, real analysis validates the logical progression from continuity at a point to continuity over an entire domain.

Functional Equations

Functional equations involve relationships between functions rather than just variables. These equations define functions by their interactions and operations.
The Cauchy functional equation \( f(x+y) = f(x) + f(y) \) is a classic example of such an equation.

• It describes a function based on the sum of its inputs without explicitly stating the form of the function.
• Solutions to such equations can be complex, but continuity conditions simplify them.

In the context of the Cauchy functional equation, a key result is that if a solution is continuous at even one point, the function must be of the linear form \( f(x) = cx \), with \( c \) being a constant. This reveals the significance of continuity in solving functional equations. Functional equations are thus a bridge between algebra and analysis, offering insight into both the structure and behavior of functions.