Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y)$, $\forall$ $x, y \in R$. If $f(x)$ is differentiable at $x=0$, then

Let f: R → R be a function such that f(x+y)=f(x)+f(y), ∀ x, y ∈ R. If f(x) is differentiable at x=0, then $f(x+y) = f(x) + f(y) \quad \forall\, x, y \in \mathbb{R}$ $f \text{ is differentiable at } x = 0$ $\text{Which of (1), (2), (3), (4) are TRUE?}$ $1.\ f(0) = 0, \ f(-x) = -f(x)$ $2.\ c := f'(0) = \lim_{h \to 0} \dfrac{f(h)}{h}, \quad \text{then } f'(x) = c \ \forall x$ $3.\ \text{judge (1)-(4)}$ $x = y = 0: \quad f(0) = 2f(0) \Rightarrow f(0) = 0$ $y = -x: \quad 0 = f(x) + f(-x) \Rightarrow f(-x) = -f(x) \ (\text{odd})$ $c := f'(0) = \lim_{h \to 0} \dfrac{f(0+h)-f(0)}{h} = \lim_{h \to 0} \dfrac{f(h)}{h}$ $\dfrac{f(x+h)-f(x)}{h} = \dfrac{f(x)+f(h)-f(x)}{h} = \dfrac{f(h)}{h} \to c \ (h \to 0)$ $\Rightarrow f'(x) = c \quad \text{for every } x \in \mathbb{R}$ $f(x) = c\,x \quad (\text{a straight line through the origin})$ $(1)\ \text{FALSE}: f \text{ is differentiable on all of } \mathbb{R}$ $(2)\ \text{TRUE (diff.} \Rightarrow \text{cont.)}, \quad (3)\ \text{TRUE } (f'(x)=c)$ $(4)\ \text{FALSE}: \text{ no exceptional point}$ $\boxed{\text{TRUE: statements } (2) \text{ and } (3)}$