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We map Lipschitz spaces and functions of bounded variation by the operator      σ  : φ  ( x ) →  ∑  0 ∞  2 −  n φ  ( 2 n x ) , x ∈  [ 0 , 1 ] \sigma :\varphi (x) \to \sum \nolimits _0^\infty {{2^{ - n}}} \varphi ({2^n}x),x \in [0,1]      , and we estimate the Hausdorff measure of      σ  ( φ  ) \sigma (\varphi )      . We furthermore introduce a class of continuous and nowhere differentiable functions on [0,1] which we call      T \mathcal {T}      . We make a refined analysis of the fractal and smoothness properties of the functions in      T \mathcal {T}       and study the relationship between the two. We show that all the functions in      T \mathcal {T}       have box dimension equal to      1 2 \frac {1}{2}      , with respect to the dimension family      { t / ( log ⁡  1 t ) s : s ∈  R + } \{ t/{(\log \frac {1}{t})^s}:s \in {\mathbb {R}^ + }\}      , but that their order of smoothness covers a wide range.

The Takaga operator, Bernoulli sequences, smoothness conditions and fractal curves