MHB Prove $f(x)=Cx$ for All $x$: Functional Equation

[alexmahone]

Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$.

I have shown that $f(x)=Cx$ for all rational numbers. How do I use the continuity of $f$ to show it is true for all $x$?

 

[Krizalid1]

I have shown that $f(x)=Cx$ for all rational numbers.

Having this, remember that rationals are dense in $\mathbb R.$

 

[alexmahone]

Having this, remember that rationals are dense in $\mathbb R.$

Intuitively, I can see that it must be true but I'm having trouble proving it.

 

[Plato2]

Intuitively, I can see that it must be true but I'm having trouble proving it.

Every real number is the limit of a sequence of rational numbers.
The function is continuous. What continuity and convergent sequences?

 

[alexmahone]

Every real number is the limit of a sequence of rational numbers.
The function is continuous. What continuity and convergent sequences?

Got it. Thanks!

 

[HallsofIvy]

By the way, if you do not include the requirement that the function be continuous, all f are either of the form f(x)= cx or the graph of y= f(x) is dense in the plane.