Let $f,g:[0,1]\to{\mathbb R}$ be two continuous functions,
Is it true that $dim_B(graph(f+g))\le \max\{\dim_B(graph(f)),dim_B(graph(g))\}$?
It is well known (in Falconer book is an exercise) that this is even an equality if both dimensions are not equal. I am specially interested in the case that both dimensions are the same. It is trivial that, in this last case, $f+g$ can be zero (just take $g=-f$) or any Lipschitz function.
So my more specific question is the next:
If $f,g:[0,1]\to{\mathbb R}$ are continuous functions whose graphs have Box-dimension $s$, ¿Is $dim_B(graph(f+g))\le s$?
Also interested, if possible, in lower and upper box-dimension.
Your first question appears as lemma 3.4 of my paper "The prevalent dimension of graphs". The proof states that this is a simple consequence of the inequality $$R_{f+g}[a,b] \leq R_f[a,b] + R_g[a,b] \leq 2\max\{R_f[a,b],R_g[a,b]\},$$ where $R_{f}[a,b] = \sup\{|f(x) - f(y)|:a<x,y<b\}$.