Originally Posted by

**DeFacto** 4. Let f : R -> R be a function satisfying

f(x + y) = f(x) + f(y) for any x, y in R.

Using elementary algebra prove that f(0) = 0 and f(−x) = −f(x) for every x;

by induction prove that f(nx) = nf(x) for every x and every natural number n;

and then show that f(rx) = rf(x) for every x and any rational number r = n/m.

These parts i have done. I'm not sure how to go about doing the next bit:

*Prove that if f is also continuous then, for some constant c,

f(x) = cx for every x in R.*

thanks.