Question

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that $$ f(x+y)=f(x)+f(y), \quad \forall x, y \in \mathbb{R} $$ If \(f(x)\) is differentiable at \(x=0\), then (A) \(f(x)\) is differentiable only in a finite interval containing zero (B) \(f(x)\) is continuous \(\forall x \in \mathbb{R}\) (C) \(f^{\prime}(x)\) is constant \(\forall x \in \mathbb{R}\) (D) \(f(x)\) is differentiable except at finitely many points

Short answer

Option (B) The function is continuous for all x in \(\mathbb{R}\). Option (C) \(f'(x)\) is constant for all x in \(\mathbb{R}\).

Step-by-step solution

Understand the functional equation

The functional equation implies that the function follows the property of additivity, meaning that the value of the function at the sum of two numbers is the sum of the values of the function at each number individually.

Analyze the differentiability at x=0

Since the function is differentiable at x=0, we have a derivative f^'(0). This implies that the limit of (f(h)-f(0))/h as h approaches 0 exists.

Determine differentiability and continuity for all x

Using the property of the function and the fact that it is differentiable at x=0, one can derive that f is both differentiable and continuous for all x in \(\mathbb{R}\). Differentiability implies continuity, thus confirming option B.

Prove that the derivative is constant

By applying the definition of derivative and using the given functional equation, one can show that \(f'(x)=f'(0)\) for all x, indicating that the derivative of the function is constant across its domain \(\mathbb{R}\), thus confirming option C.

Rule out other options

As we have shown that options B and C are correct, it is clear that options A and D must be incorrect. Option A asserts that the function is differentiable only near zero, while option D suggests it is not differentiable at certain points; both are inconsistent with our findings.

Key concepts

Additive Functions

An additive function adheres to a simple yet profound property: if you have two inputs, say, x and y, and you sum them up, the resultant value is the same as if you applied the function individually to each input and then summed those results. This can be mathematically represented as
\[ f(x + y) = f(x) + f(y) \]
for all real numbers x and y. Additive functions showcase a kind of 'building block' behavior, where the whole is truly the sum of its parts. This characteristic often simplifies complex problems, especially in algebra and functional analysis.

Derivative Continuity

The continuity of a derivative is a deep concept in calculus. It's related to how smoothly a function changes. If a function is differentiable at every point in its domain, its derivative is a new function that tells you the rate of change at each point. Now, if this derivative function doesn't make any sudden jumps or breaks, we say it's continuous.

Derivative continuity is crucial because it often ensures that the function is well-behaved and predictable across its entire domain. When you have derivative continuity, you can use all sorts of calculus tools more effectively, like integration.

Constant Derivative

When we say a function has a constant derivative, we mean that no matter where you look on the function's graph, the slope or rate of change is always the same. This is a fascinating property because it implies that the original function must be a straight line.

Mathematically, if for all x in the domain of the function, we find that \[ f'(x) = c \] for some constant c, then our function marches forward at a steady rate, never speeding up or slowing down, like a car cruising at a constant speed on a highway with no turns.

Functional Equation

A functional equation is like a puzzle or a riddle that involves functions. It's an equation where the variables you're solving for are entire functions, not just numbers.

These equations can take on many forms, but they all describe relationships between functions and their values. Functional equations give critical insights into the possible behaviors of functions and are a cornerstone of understanding complicated mathematical relationships. They are indispensable in various areas like number theory, analysis, and mathematical physics.

Differentiability

Differentiability is a quality of smoothness for functions. If a function is differentiable at a point, it means that it has a tangent line at that point, and thus, a well-defined slope. In essence, you can zoom in as close as you like to the curve of the function at this point, and it will eventually start to look like a straight line.

Differentiability across an entire function implies a certain smoothness and regularity, which often simplifies both theoretical and practical applications. If a function is differentiable, then it's also continuous, but the reverse doesn't always hold, making differentiability a stronger condition than continuity.