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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.

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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](http://homepages.math.uic.edu/~marker/math414/fs.pdf) might be of interest to you. – 2017-02-11

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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 $01+\frac{3\pi}{2}$, 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. – 2017-02-11
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    Thanks for the example. I am familiar with the Weierstraß function. I am looking for a class of formal definitions. – 2017-02-11
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    @Stahl Thank you. I added it! – 2017-02-11