The discussion revolves around proving that a function \( f(x) \) satisfying the functional equation \( f(a+b)=f(a)+f(b) \) for all \( a \) and \( b \), and being continuous, must take the form \( f(x)=Cx \) for all \( x \). The scope includes theoretical exploration and mathematical reasoning regarding the properties of continuous functions and their implications.
Participants generally agree on the form \( f(x)=Cx \) for rational numbers and the importance of continuity in extending this to all real numbers. However, the exact proof and the role of continuity remain points of exploration and are not fully resolved.
Limitations include the need for a rigorous proof that connects the behavior of \( f \) on rational numbers to its behavior on real numbers, particularly through the use of continuity and limits.
Having this, remember that rationals are dense in $\mathbb R.$Alexmahone said:I have shown that $f(x)=Cx$ for all rational numbers.
Krizalid said:Having this, remember that rationals are dense in $\mathbb R.$
Every real number is the limit of a sequence of rational numbers.Alexmahone said:Intuitively, I can see that it must be true but I'm having trouble proving it.
Plato said:Every real number is the limit of a sequence of rational numbers.
The function is continuous. What continuity and convergent sequences?