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Is there a formal definition for the concept of nowhere differentiable function in $\mathbb{R}$?

I know that a function can fail to be differentiable in several ways in an open interval but I am looking for references. I am also familiar with the most common examples.

user48672
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    It is a function $f : \Bbb R\to\Bbb R$ such that there is no $a\in\Bbb R$ where the limit $\lim_{x\to a}\frac{f(x) - f(a)}{x - a}$ exists. An easy class of examples is the set of functions which are nowhere continuous. This might be of interest to you. – Stahl Feb 11 '17 at 00:10
  • This was recently bumped up to the main MSE page by "Community Bot" for some reason, so I guess I'll make a comment for those desiring more information about this topic. The usual meaning is a finite derivative not existing at each point of its domain (typically an interval), but sometimes (probably only in relatively old literature) a stronger notion is implied in which neither a finite nor an (signed) infinite derivative exists at each point (e.g. see my mathoverflow answers 1 & 2). (continued) – Dave L. Renfro Feb 23 '25 at 20:25
  • Also, there are many variations on the notion of differentiable (e.g. see this MSE answer. Finally, there is a fairly large literature on various ways in which "most" continuous functions are nowhere differentiable (in various ways) -- see this MSE answer for a brief overview of some results of this nature. – Dave L. Renfro Feb 23 '25 at 20:25

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Such a function exists but it is nontrivial to show that it is nowhere differentiable. The Weierstraß function $$W:\mathbf R \to \mathbf R, \quad W(x)=\sum_{k\in\mathbf N}b^k \cos (a^k x)$$ for $0<b<1$ and $ab>1+\frac{3\pi}{2}$, where $a\in\mathbf N$, $a\neq 0$ and $b\in\mathbf R$ is continuous but nowhere differentiable. The idea to show this is by contradiction, using fourier methods and some growing conditions. If you like I can post the idea later in more detail.

The picture shows the partial sum of $W$ with $\alpha= 0.5$

Partial sum of $W$ with $\alpha= 0.5$.

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    It's trivial to obtain a nowhere differentiable function (say, a function which is $0$ on $\Bbb Q$ and $1$ on $\Bbb R\setminus\Bbb Q$), but it is nontrivial to construct a continuous nowhere differentiable function (which is what the Weierstraß function is. – Stahl Feb 11 '17 at 00:07
  • Thanks for the example. I am familiar with the Weierstraß function. I am looking for a class of formal definitions. – user48672 Feb 11 '17 at 00:08