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The geometry of the multifractional Brownian motion (mBm) is known to presenta complex and surprising form when the Hurst function is greatly irregular.Nevertheless, most of the literature devoted to the subject considerssufficiently smooth cases which lead to sample paths locally similar to afractional Brownian motion (fBm). The main goal of this paper is therefore toextend these results to a more general frame and consider any type ofcontinuous Hurst function. More specifically, we mainly focus on obtaining acomplete characterization of the pointwise H\"older regularity of the samplepaths, and the Box and Hausdorff dimensions of the graph. These results, whichare somehow unusual for a Gaussian process, are illustrated by severalexamples, presenting in this way different aspects of the geometry of the mBmwith irregular Hurst functions

Some sample path properties of multifractional Brownian motion