Hausdorff Dimension for Brownian motion over [0,1] I am trying to calculate Hausdorff dimension for the trajectory of Brownian motion over $[0,1]$. I read the book of Morters and Peres and know that the dimension will be $\frac{3}{2}$. I tried to use different approach from the book and I have some concerns as follow:


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*Is it true that: If $f$ is a Lipschitz function or has a continuous derivative or has bounded variation, then graph $f $ has dimension of $1$. If this is true, can someone please explain to me or point me to some relevant material.

*I calculate the total variation of Brownian motion (BM) and got the result as follow:
$$\sum_{i=1}^{n} |B(t_i)-B(t_{i-1})| \longrightarrow \sqrt{\frac{2}{\pi h}}$$
this shows that Brownian motion has unbounded variation as $h \to 0$ $(h=t_i-t_{i-1})$. If 1. is true  then this implies 1-dimensional Brownian motion has dimension of greater than $1$. If I look at the total variation of BM, the total variation will become larger as $h$ become smaller and the order of magnitude of the sum is $\frac{1}{2}$ (the exponent of $h$). I wonder if this is the Hausdorff dimension of BM because it is not $\frac{3}{2}$ as mentioned in a lot materials.
I really appreciate any helps or comments 

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*Every continuous function of bounded variation (BV) has a graph of Hausdorff dimension $1$. Indeed, being BV means precisely that the graph is rectifiable, i.e., has finite $1$-dimensional Hausdorff measure. Being Lipschitz continuous implies BV. Having continuous derivative implies Lipschitz continuity, hence implies BV.  

*It's true that sample paths of Brownian motion have infinite variation. By itself, infinite variation does not imply Hdim $>1$. One has to look more carefully at the sample path (quadratic variation, modulus of continuity...) to conclude that Hdim $=3/2$ for almost every path. This is not an easy result; it was proved in 1953 by S.J. Taylor. A proof can be found in An Invitation to Sample Paths of Brownian Motion by Yuval Peres; see Theorem 6.1.
