1. We are given the functional equation

   \,\forall x,y\in \mathbb R.$$ This is Cauchy’s functional equation.

2. Also given: $f$ is differentiable at $x=0$.

3. Since differentiability at $0$ implies continuity at $0$, we have $\lim_{h\to 0} f(h)=f(0).$

4. First find $f(0)$ using the functional equation: $f(0)=f(0+0)=f(0)+f(0)\implies f(0)=0.$

5. Because $f$ is continuous at $0$, and additive, it must be linear. Let us prove it using differentiability at $0$.

6. Differentiate at $0$: $f'(0)=\lim_{h\to 0}\frac{f(h)-f(0)}{h}=\lim_{h\to 0}\frac{f(h)}{h}.$ Let $f'(0)=c.$ Then for small $h$, $\frac{f(h)}{h}\to c.$

7. Now check differentiability at any arbitrary $x\in\mathbb R$: $f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.$ Using additivity, $f(x+h)-f(x)=f(h).$ Therefore, $f'(x)=\lim_{h\to 0}\frac{f(h)}{h}=f'(0)=c.$ Hence $f'(x)$ exists for every $x\in\mathbb R$ and is the same constant.

8. So:

   • $f$ is differentiable for all $x\in\mathbb R$.
   • $f'(x)$ is constant for all $x\in\mathbb R$.

9. Since differentiability implies continuity, $f$ is continuous for all $x\in\mathbb R$.

10. Now evaluate the options:

• A: "Differentiable only in a finite interval containing zero" — false, since it is differentiable for all real $x$.
• B: "$f(x)$ is continuous $\forall x\in\mathbb R$" — true.
• C: "$f'(x)$ is constant $\forall x\in\mathbb R$" — true.
• D: "Differentiable except at finitely many points" — false, since it is differentiable everywhere.

Therefore the correct options are $\boxed{B,\ C}.$