If a function $f:R\to R$ satisfy the equation $f(x+y)=f\left(x\right)+f\left(y\right),\forall x,y$ and the function $f\left(x\right)$ is continuous at $x=0$, then

• A

  $f\left(x\right)$ is continuous for all positive real values of $x$
• Correct answer
  B

  $f\left(x\right)$ is continuous for all $x$
• C

  $f\left(x\right)=0$ for all $x$
• D

  None of the above

Hence, the correct answer is option: 'B'.