Question

Let $f(x+y)=f\left(x\right)+f\left(y\right)$ for all x and y. If the function $f\left(x\right)$ is continuous at $x=0$, show that $f\left(x\right)$ is continuous for all x.

Answer 1

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

Let $x_1$, and $x_2$ be the two variables,

$f(x_1+x_2)=f\left(x_1\right)+f\left(x_2\right)$

So, $f\left(x\right)$ is continuous for all $x$.