The discussion centers on proving that a continuous function \( f(x) \) satisfying the functional equation \( f(a+b) = f(a) + f(b) \) for all \( a \) and \( b \) must take the form \( f(x) = Cx \), where \( C = f(1) \). The proof is established for rational numbers, leveraging the density of rationals in the real numbers and the continuity of \( f \) to extend the result to all real numbers. The continuity condition is crucial, as it differentiates between functions that are linear and those that may be dense in the plane without continuity.
PREREQUISITESMathematicians, students of real analysis, and anyone interested in functional equations and their properties will benefit from this discussion.
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?