Dimension of a graph

Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?

If not, what extra condition do we need? (Something about the "size" of its "jumps", possibly?)

• The characteristic function of the rationals is discontinuous everywhere, but its graph is contained in the union of two line segments, so box dimension is 1. Dec 29, 2016 at 13:15
• Indeed. So, we do need something extra to discard such obvious counterexamples. Dec 29, 2016 at 13:42
• To generalize Vaughn's comment, let $f:[0,1]\to [\delta,\infty)$ be any function, and let $g$ agree with $f$ on the irrationals and be $0$ on the rationals. Then the lower box dimension of the graph of $g$ is smaller or equal than that of $f$, with equality if $f$ is continuous. So it seems to me that being discontinuous at all points, regardless of the size of jumps, in itself has no bearing on the box dimension of the graph. Dec 29, 2016 at 14:25
• To avoid this kind of constructions, one would need to assume something far stronger of the type: if $f|_D$ is continuous, then $D$ is nowhere dense. I have no intuition for what kind of functions satisfy this, but they might be wild enough for their graph to have large (even full?) box dimension. Dec 29, 2016 at 14:34