The problem involves a function f: R → R that satisfies the functional equation f(x+y) = f(x) + f(y) for all real numbers x and y. The original poster seeks to demonstrate that if f is continuous, then there exists a real number b such that f(x) = bx.
Some participants have offered hints and partial reasoning, particularly regarding the relationship between f(x) and f(1). There is an ongoing exploration of how to generalize findings from rational numbers to all real numbers, indicating a productive direction in the discussion.
Participants are navigating the constraints of continuity and the specific properties of the function as defined by the problem statement. There is a focus on the implications of the functional equation and the continuity condition without resolving the overall question.