Question

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

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

Answer 1

The correct option is C $200$

From question, we know that:

$f(x+y)=f\left(x\right)+f\left(y\right)$

Let $x=1,y=1$

$f\left(2\right)=2f\left(1\right)=4=2\times 2$

$f\left(4\right)=2f\left(2\right)=8=2\times 4$ when $(x=2,y=2)$

$f\left(8\right)=2f\left(4\right)=16=2\times 8$ when $(x=4,y=4)$

$f\left(16\right)=2f\left(8\right)=32=2\times 16$ when $(x=8,y=8)$

Hence, it can be generalized that $f\left(x\right)=2x$

$\therefore f\left(100\right)=2\times 100=200$