Space of Differentiable Functions I am writing and I want to define a space of differentiable functions. I am aware of the space $C^1[a,b]$ however this also puts some constraints on the regularity of $f^\prime$. In particular, $f^\prime\in C^1$ if $f$ is continuous, differentiable, and has continuous derivative. Is there some standard space that is used to denote the space of differentiable functions that doesn't necessarily require $f^\prime$ to be continuous?
 A: Your instincts are correct.  If you wish to study derivatives (or differentiable functions which is the same thing essentially) you are well-advised to introduce  appropriate function spaces.

Analogy:  Does there exist a continuous function $f:[a,b]\to\mathbb R$ such that $f$ is nowhere monotonic (i.e., it is
not monotonic on any subinterval of $[a,b]$)?

The 19th century answer was provided by a number of explicit constructions of nowhere differentiable functions, all of which are (of course) nowhere monotone.  But the more modern approach is via function spaces.  In 1931 the Polish mathematicians Mazurkiewicz and Banach had initiated the study of "typical" properties of continuous functions.   Introduce the space $\cal C[a,b]$ of continuous functions equipped with the sup norm.  A property is typical [or generic] if the subset of functions in $\cal C[a,b]$  with that property is residual in the Banach space  $\cal C[a,b]$.
So the modern answer to the problem posed is that, not merely is there a single example of a nowhere monotonic continuous function:  such functions are typical in that "most" continuous functions have that property.  There is a huge literature devoted to typical properties of continuous functions.

Problem:  Does there exist a differentiable function $f:[a,b]\to\mathbb R$ such that $f$ is nowhere monotonic (i.e., it is
not monotonic on any subinterval of $[a,b]$)?

The Roumainian mathematician Pompeiu constructed such a function in 1907.  His construction had some problems and over the years there were other simpler and more correct constructions given (e.g., [3], [4]).
But why can't we do the same thing as before and put differentiable functions into an appropriate Banach space and use category arguments.  [I think that is what the OP might ask here?]
Well yes we can.  Cliff Weil [2] did exactly that and showed that, in the appropriate space, functions of the Pompeiu type (differentiable, nowhere monotone) are typical.
Details.  Consider the following collections of functions:

*

*$\cal bC_{ap}$ of bounded approximately continuous function on $[a,b]$.


*$\cal b\Delta'$ of bounded derivatives on $[a,b]$.


*$\cal bDB_1$ of bounded Darboux Baire 1 functions on $[a,b]$.


*$\cal bB_1$ of bounded Baire 1 functions on $[a,b]$.
Note that
$$\cal bC_{ap} \subset  \cal b\Delta'\subset  \cal bDB_1 \subset \cal bB_1$$
and that each family is closed under uniform limits.  Thus each one is a Banach space when furnished with the sup norm.  Each one is a closed, nowhere dense subset of the next larger space.
These Banach spaces have been extensively studied and play an important role in the investigation of derivatives.  The space $b\Delta'$ is the one used in Cliff Weil's study of Pompeiu functions.  Chapter 15 of Andy Bruckner's monograph [5] will introduce you to those ideas and to some of the {large} literature of these interesting function spaces.
REFERENCES:
[1] Pompeiu, Dimitrie (1907). "Sur les fonctions dérivées". Mathematische Annalen. 63 (3): 326–332.
[2] Weil, Clifford E. On nowhere monotone functions. Proc. Amer. Math. Soc. 56 (1976), 388–389.
[3]  Casper Goffman, Everywhere differentiable functions and the density topology, Proc. Amer. Math. Soc. 51 (1975), 250.
[4] Y. Katznelson and K. Stromberg, Everywhere differentiable, nowhere monotone functions, Amer. Math. Monthly 81 (1974), 349-354.
[5] Bruckner, Andrew.  Differentiation of real functions.
Second edition. CRM Monograph Series, 5. American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6
