A good reference for this is the paper

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. (Details of the construction can be found here.)

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.)

For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.

A good reference for this is the paper

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. (Details of the construction can be found here.)

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.)

For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.

A good reference for this is the paper

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval.

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.)

For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.

A good reference for this is the paper

Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353.

There, the authors give an explicit construction of a function $L:\mathbb R\to\mathbb R$ that is differentiable, with bounded derivative, and such that all three sets $\{x\mid L'(x)>0\}$, $\{x\mid L'(x)=0\}$, and $\{x\mid L'(x)<0\}$ are dense in $\mathbb R$. The argument is flexible enough that we can arrange $\{x\mid L'(x)=0\}=\mathbb Q$, if wanted. The function $L$ is the difference of two monotone functions, and $L'$ is not Riemannn integrable over any interval. (Details of the construction can be found here.)

In the paper, they indicate that the first example of a differentiable, nowhere monotone, function is due to Köpke, in 1889, with a further example due to Pereno in 1897. (I have not examined their constructions myself, so I do not know whether there are mistakes in their presentation, as the other answer suggests, but Pereno's function is discussed as an example in Hobson's book from 1957.)

For this kind of results, two obvious references to consult are Counterexamples in Analysis, by Gelbaum and Olmsted, and A second course on real functions, by van Rooij and Schikhof. The latter presents the Katznelson-Stromberg construction in detail.