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

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

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

Step 1: Understanding the Functional Equation

The given functional equation is \( f(x+y) = f(x) + f(y) \). This is known as Cauchy's Functional Equation. It holds for all real numbers \( x \) and \( y \).

Step 2: Property of the Zero Function

A trivial solution to Cauchy's Functional Equation is \( f(x) = 0 \) for all \( x \). This function is certainly continuous everywhere, including \( x=0 \).

Step 3: Non-Trivial Solutions and Continuity

For the stated functional equation, non-trivial (i.e., non-linear) solutions exist only under certain conditions. However, if we require that \( f(x) \) be continuous at even a single point, this implies \( f(x) \) must be of the form \( f(x) = cx \) for some constant \( c \). This is because the only continuous solutions of Cauchy's equation are linear.

Step 4: Application of Continuity at \(x = 0\)

Given that \( f(x) \) is continuous at \( x = 0 \), it means \( f(x) = cx \) where \( c \) is a constant. This function is continuous everywhere because it's a linear function. Thus, \( f(x) \) is continuous for all \( x \).

Step 5: Conclusion

Based on the analysis, the function \( f(x) = cx \) satisfies both the functional equation and the continuity condition, indicating that \( f(x) \) is continuous for all real numbers.

Key Concepts

Continuity at a Point

In mathematics, continuity of a function at a particular point means that the function behaves well around that point. For a function \( f(x) \), being continuous at a point \( x = a \) means the limit as \( x \) approaches \( a \) from either side is equal to the function’s value at \( a \). In equation form: \[\lim_{{x \to a}} f(x) = f(a)\]When it comes to Cauchy's Functional Equation, continuity at a single point can vastly influence the nature of solutions. Specifically, if a function \( f(x) \) is continuous at even one point, it must take a specific form. In the case provided, continuity at \( x=0 \) implies that \( f(x) \) must be linear in nature. This is because any deviation from linearity would lead to discontinuities. Thus, stating that the function is continuous at one point can tell us a lot about its global behavior.

Linear Functions

Linear functions are one of the simplest and most fundamental types of functions in mathematics. They are defined by the formula \( f(x) = mx + b \), where \( m \) and \( b \) are constants. Such functions graph as straight lines. A unique property of linear functions is that they maintain a constant rate of change - the slope \( m \).
In the context of Cauchy's Functional Equation, when a function is continuous at a point and satisfies \( f(x+y) = f(x) + f(y) \), it implies \( f(x) \) must take the form \( f(x) = cx \) for some constant \( c \). This is because linear functions are consistent with both the equation and continuity requirements. Linear functions are tremendously important as they make up the backbone of many mathematical models and real-world applications, providing simplicity and predictability in their behavior.

Real Numbers

Real numbers are the set of all numbers that can be found on the number line. This includes all rational numbers, such as fractions and integers, as well as all irrational numbers, which can't be expressed as fractions. Real numbers are fundamental in continuous mathematics, which heavily relies on the notion of small changes.
When dealing with problems involving Cauchy's Functional Equation, it applies to all real numbers \( x \) and \( y \). This universality in the domain means that whatever properties we derive, such as continuity or linearity, hold across the entire set of real numbers. This consistency across the entire domain allows us to make generalized conclusions about the nature of the function \( f(x) \) based solely on its behavior at a single point. Thus, the use of real numbers in mathematical problems enables us to apply techniques like differentiation and integration, aiding in the exploration of continuous functions.