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W. Rudin gives the following example $f(x)$.

Let $\varphi(x) = |x|$ on $[-1,1]$ and extend periodically to the whole real line with period $2$.

Now define

$$ f(x) = \sum_{n=0}^\infty \left(\frac{3}{4}\right)^n\varphi(4^nx).$$

How does the graph of this function, or its successive approximations, look like? Just to get an idea of how continuity happens, but not differentiability.

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    I suspect that $\varphi$ is defined on $[-1,1]$, and that the function is supposed to be $$f(x)=\sum_{n=0}^\infty\left(\frac34\right)^n\varphi(4^nx)\;.$$2012-06-08

2 Answers 2