Question

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

• A. $f\left(x\right)=0$ continuous for all $x$
• B. $f\left(x\right)$ is continuous for all positive real $x$
• C. $f\left(x\right)$ is continuous for all $x$
• D. None of these

Answer 1

The correct option is C $f\left(x\right)$ is continuous for all $x$

Since $f\left(x\right)$ is continuous at $x=0,$

$\therefore \underset{x\to 0}{lim}f\left(x\right)=f\left(0\right)$

Take any point $x=a,$ then at $x=a$

$\underset{x\to a}{lim}f\left(x\right)=\underset{h\to 0}{lim}f(a+h)=\underset{h\to 0}{lim}[f\left(a\right)+f\left(h\right)]$

$=f\left(a\right)+\underset{h\to 0}{lim}f\left(h\right)=f\left(a\right)+f\left(0\right)=f(a+0)=f\left(a\right)$

$\therefore f\left(x\right)$ is continuous at $x=a.$

Since $x=a$ is any arbitrary point, therefore $f\left(x\right)$ is continuous for all $x.$