Question

If $f:R\to R$ is continuous such that $f(x+y)=f\left(x\right)+f\left(y\right),\forall x\in R,y\in R$ and $f\left(1\right)=2$ then $f\left(100\right)=$

• A. $100$
• B. $50$
• C. $200$
• D. $0$

Answer 1

The correct option is C $200$

The general function f satisfying $f(x+y)=f\left(x\right)+f\left(y\right)$ will be linear.
$\Rightarrow f\left(x\right)=ax+b$
$f(x+y)=f\left(x\right)+f\left(y\right)$
$\Rightarrow b=0$
$\therefore f\left(x\right)=ax$
$f\left(1\right)=2\Rightarrow a=2$
$\therefore f$ is $f\left(x\right)=2x$
$f\left(100\right)=200$