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In many books from mathematical analysis (for example in Rudin) is presented the following example of continuous nowhere differentiable function:

$$f(x)=\sum_{n=1}^\infty (\frac{3}{4})^n g(4^n x) \textrm{ for } x \in \mathbb{R},$$

where $g(x)=|x|$ for $x \in [-1,1]$ and $g(x+2)=g(x)$ for $x \in \mathbb{R}$.

Who found this example for the first time?

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It is a simple variation of Karl Weierstrass's function, which uses cosines rather than periodic absolute values – Henry Aug 17 '12 at 8:13
    
Thinks, I know but I look for author or only for name of this particular function. – R.S Aug 17 '12 at 8:18
2  
I think it's what is usually called blancmange curve or Takagi function. Wikipedia informs me that it's also called the Takagi-Landsberg function. – t.b. Aug 17 '12 at 8:42
    
@t.b. I would reserve blancmange for the critical case $\alpha\beta=1$ (when the graph has Hausdorff dimension 1). Otherwise the visual resemblance to pudding isn't there. I think this was also the case studied by Takagi originally, but I'm not sure. In any event, the supercritical case $\alpha\beta>1$ does not require anything that Weierstrass did not know (for the proof of nondifferentiability). – user31373 Aug 18 '12 at 4:58

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