Properties of Additive Cauchy Functional Equation with Differentiability at Zero

To determine the correct options, we analyze the given Cauchy functional equation:
$f(x + y) = f(x) + f(y), \quad \forall x, y \in \mathbb{R}$

Step 1: Finding $f(0)$

Substitute $x = 0$ and $y = 0$ into the equation:
$f(0 + 0) = f(0) + f(0) \implies f(0) = 2f(0) \implies f(0) = 0$

Step 2: Differentiability at $x = 0$

It is given that $f(x)$ is differentiable at $x = 0$. By the definition of the derivative:
$f'(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0} \frac{f(h)}{h}$
Let $f'(0) = k$, where $k$ is a finite real constant.

Step 3: Differentiability and Derivative at any Arbitrary $x \in \mathbb{R}$

Using the definition of the derivative for an arbitrary $x \in \mathbb{R}$:
$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$

Using the given functional relation $f(x + h) = f(x) + f(h)$:
$f'(x) = \lim_{h \to 0} \frac{f(x) + f(h) - f(x)}{h} = \lim_{h \to 0} \frac{f(h)}{h}$

Since $\lim_{h \to 0} \frac{f(h)}{h} = f'(0) = k$, we have:
$f'(x) = k \quad \forall x \in \mathbb{R}$

Thus:

• The derivative $f'(x)$ exists for all $x \in \mathbb{R}$, meaning $f(x)$ is differentiable everywhere on $\mathbb{R}$.
• $f'(x)$ is a constant for all $x \in \mathbb{R}$. Hence, Option (C) is correct.

Step 4: Continuity of $f(x)$

Since differentiability on an interval or domain implies continuity everywhere on that domain, and $f(x)$ is differentiable for all $x \in \mathbb{R}$, it follows directly that $f(x)$ is continuous for all $x \in \mathbb{R}$.
Alternatively, for any $x \in \mathbb{R}$:
$\lim_{h \to 0} [f(x + h) - f(x)] = \lim_{h \to 0} f(h) = \lim_{h \to 0} \left( h \cdot \frac{f(h)}{h} \right) = 0 \cdot k = 0$
which proves continuity. Hence, Option (B) is correct.

Step 5: Analysis of Option (A) and Option (D)

• Option (A) states that $f(x)$ is differentiable only in a finite interval containing zero. Since $f(x)$ is differentiable on all of $\mathbb{R}$, this statement is false.
• Option (D) states that $f(x)$ is differentiable except at finitely many points. Since $f(x)$ is differentiable at every point in $\mathbb{R}$, the set of points where it is non-differentiable is the empty set $\emptyset$. As the empty set has cardinality $0$ (which is finite), the statement is mathematically vacuously true.

Therefore, the correct options are B, C, and D (with B, C also accepted based on standard linguistic interpretation).