How to interpret the functional identity $f(a+b)=f(a)+f(b)$?

Question

I know that $y=kx$ is definitely a solution. Other solutions may be constructed by treating the real numbers as a vector field over the rational numbers, which are pathological.

Question: Is there such an $f$ having its graph dense in $\mathbb{R}^2$?

see here https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation — Dr. Sonnhard Graubner, Jan 01 '18 at 15:11

score 3 · Accepted Answer · answered Jan 01 '18 at 15:09

3

All non-linear (or "pathological") solutions have graphs that are dense in the plane. See, eg, this SEM overview article. Or this one.

answered Jan 01 '18 at 15:09

kimchi lover

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How to interpret the functional identity $f(a+b)=f(a)+f(b)$?

1 Answers1