This [Hausdorff dimension of the graph of an increasing function](https://mathoverflow.net/questions/304573/hausdorff-dimension-of-the-graph-of-an-increasing-function) shows that:



> 
> Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
>  itself with $f(0)=0, f(1)=1$. Then $dim\_H \; G = 1$ where $G$ is the graph of $f$. 
> 
> 
> 


I have at hand the Casino function, described as follows in Massopoust's *Interpolation and Approximation with Splines and Fractals*:



> 
> Let $X = [0,1] \times \mathbb{R}$, $N = 4$ and $Y = \{(x\_v,y\_v):0 = x\_0 < \ldots x\_N = 1, 0 = y\_0 < \ldots < y\_N = 1\}$. Define an IFS by $f\_i(x,y) = 
> \begin{pmatrix}
> x\_i-x\_{i-1} & 0 \\
> 0 & y\_i - y\_{i-1}
> \end{pmatrix}
> \begin{pmatrix}
> x \\
> y
> \end{pmatrix}
> +
> \begin{pmatrix}
> x\_{i-1} \\
> y\_{i-1}
> \end{pmatrix}
> $ for $i = 1, \ldots, N$.
> 
> 
> The associated RB operator $T$ is contractive and its unique fixed point is called a Casino function $c:[0,1] \to [0,1]$. These functions are monotone increasing and therfore $dim\_H \; graph(c) = \dim\_B \; graph(c) = 1$.
> 
> 
> 


I was wondering how can I show that $dim\_B \; graph(c) = 1$ and whether there is a general argument establishing:



> 
> Let $f$ be a continuous, strictly increasing function from $[0,1]$ to
>  itself with $f(0)=0, f(1)=1$. Then $dim\_B \; G = 1$ where $G$ is the graph of $f$. 
> 
> 
> 


I don't find an argument stablishing $dim\_B \; G \le 1$.



Box dimension of the graph of an increasing function
[Pietro Majer's argument that you cited](https://mathoverflow.net/questions/304573) actually shows that the upper box dimension is $1,$ and hence the lower box, the upper and lower packing, and the Hausdorff dimensions are all equal to $1.$ Also, graphs of real-valued Lipschitz functions defined on an interval (of positive length) have upper box dimension $1,$ and as Pietro Majer also pointed out there in a comment, the graph of a strictly increasing continuous function is geometrically congruent to the graph of a Lipschitz function.


Regarding graphs of Lipschitz functions, see Corollary 11.2(a) [using $s=1$] on p. 147 of Falconer's 1990 book [**Fractal Geometry**](https://rads.stackoverflow.com/amzn/click/com/0471922870). Regarding graphs of monotone and bounded variation functions, see p. 120 (beginning of Section 10.4) and Section 12.4 (pp. 148-150) of Tricot's 1995 book [**Curves and Fractal Dimension**](https://rads.stackoverflow.com/amzn/click/com/0387940952) and [this paper](https://www.sciencedirect.com/science/article/pii/S0362546X10000842?via%3Dihub). 


