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?
Space of Differentiable Functions
analysisderivativesfunctional-analysisfunctionsreal-analysis
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This post serves to elaborate on the comments made by copper.hat and Branimir Ćaćić above. In what follows, by a continuous function on a topological space, we shall mean an $ \mathbb{R} $- or $ \mathbb{C} $-valued continuous function on the space.
Part 1
The space of continuous functions is an interesting object of study. Indeed, we have a complete classification of commutative C*-algebras in terms of such spaces. This classification result is known as the Commutative Gelfand-Naimark Theorem, and its precise statement is as follows.
Theorem 1 (Gelfand-Naimark) Let $ \mathcal{A} $ be a commutative C*-algebra. If $ \mathcal{A} $ is unital (i.e. it has an identity element), then $ \mathcal{A} $ is isometrically *-isomorphic to $ C(X,\mathbb{C}) $ for some compact Hausdorff space $ X $. If $ \mathcal{A} $ is non-unital, then $ \mathcal{A} $ is isometrically *-isomorphic to $ {C_{0}}(X,\mathbb{C}) $ for some non-compact, locally compact Hausdorff space $ X $.
Hence, C*-algebras, which are abstract algebro-topological objects, can be concretely realized as algebras of continuous $ \mathbb{C} $-valued functions on locally compact Hausdorff spaces.
We also have the following result stating that compact Hausdorff spaces are classified, up to homeomorphism, by their ring of continuous $ \mathbb{R} $-valued functions.
Theorem 2 (Gelfand-Kolmogorov) If $ X $ and $ Y $ are compact Hausdorff spaces, then $ X $ and $ Y $ are homeomorphic if and only if $ C(X,\mathbb{R}) $ and $ C(Y,\mathbb{R}) $ are ring-isomorphic.
Yet another strong classification result is the Serre-Swan Theorem, which says that for compact Hausdorff spaces $ X $, there is an equivalence of categories between
the category of $ \mathbb{R} $- or $ \mathbb{C} $-vector bundles on $ X $ and
the category of finitely-generated projective modules over (resp.) $ C(X,\mathbb{R}) $ or $ C(X,\mathbb{C}) $.
Hopefully, the array of results presented here will imbue the reader with a deeper appreciation of the importance of spaces of continuous functions.
Part 2
Being important and being computationally useful are two different things, which is what the OP may have had in mind. This is especially so when we consider topological spaces that are equipped with a differential structure, which makes them differentiable manifolds. If we only use the space of continuous $ \mathbb{R} $-valued functions to study these manifolds, then we are doing ourselves a great disservice by failing to exploit their differential structure, which can be rich in topological information. Before giving a simple illustration, let us provide a definition.
Definition Let $ X $ be a topological space and $ \mathcal{A} $ an algebra of $ \mathbb{R} $-valued functions on $ X $. Given an $ x \in X $, we say that a mapping $ \delta: \mathcal{A} \to \mathbb{R} $ is a point-derivation on $ \mathcal{A} $ at $ x $ if $ \delta $ is an $ \mathbb{R} $-linear homomorphism and $$ \forall f,g \in \mathcal{A}: \quad \delta(fg) = f(x) \cdot \delta(g) + \delta(f) \cdot g(x). $$ The set of all point-derivations on $ \mathcal{A} $ at $ x $, which happens to form an $ \mathbb{R} $-vector space, is denoted by $ {\text{Der}_{x}}(\mathcal{A}) $.
Let $ X $ be any topological space. Then $ C(X,\mathbb{R}) $ is an algebra of $ \mathbb{R} $-valued functions on $ X $. We can therefore ask ourselves: What is $ {\text{Der}_{x}}(C(X,\mathbb{R})) $ for a given $ x \in X $? Well, it turns out that $$ \forall x \in X: \quad {\text{Der}_{x}}(C(X,\mathbb{R})) = \{ 0_{C(X,\mathbb{R}) \to \mathbb{R}} \}, $$ which is just the trivial vector space! Not very interesting indeed. (Click here to see a proof.)
Suppose this time that $ X $ is a differentiable $ n $-dimensional manifold. Then $ {C^{1}}(X,\mathbb{R}) $ is also an algebra of $ \mathbb{R} $-valued functions on $ X $. However, $ {\text{Der}_{x}}({C^{1}}(X,\mathbb{R})) $ will be an $ n $-dimensional vector space for any $ x \in X $ (the previous link also contains an explanation of this).
Note: The point-derivations on $ {C^{1}}(X,\mathbb{R}) $ at $ x $ can be used to define the tangent space of $ X $ at $ x $.
In summary, for a differentiable manifold $ X $, the vector space of point-derivations on $ C(X,\mathbb{R}) $ at any point gives us no information whatsoever. However, by considering the algebra of differentiable functions on $ X $ instead, this very same vector space can tell us the dimension of the manifold, which is an important topological invariant.
Morse Theory takes this concept to a new level of sophistication by using smooth functions on a smooth manifold to extract even more topological information. As an example, let us take a look at the following result, which is one of the jewels of Morse Theory.
Theorem 3 Let $ M $ be a compact smooth $ n $-dimensional manifold, and suppose that there exists a smooth $ \mathbb{R} $-valued function on $ M $ having exactly two critical points, both of which are non-degenerate. Then $ M $ is homeomorphic to $ \mathbb{S}^{n} $.
This result is truly amazing, for it gives us a criterion by which the differential structure of a compact smooth manifold can actually determine its topology. One can also use smooth functions to construct a new homology theory for smooth manifolds, called Morse homology, which turns out to be isomorphic to singular homology.
Many deep theorems can be proven using Morse Theory, such as the Bott Periodicity Theorem for the homotopy groups of classical Lie groups and the existence of exotic smooth structures on $ \mathbb{S}^{7} $ (a result for which John Milnor was awarded the Fields Medal).
Conclusion: The continuous functions on a topological space are important, but if you are given a differential structure, then you might want to exploit this structure by studying the differentiable functions instead because they can provide you with valuable topological information.
This answer actually answers your question, too. (See also point (2) here.)
Yes. The antiderivative of an integrable function is absolutely continuous. If $f$ is $C^1$ and of bounded variation, then $\int \lvert f'\rvert = V(f) < \infty$. So $f$ is the antiderivative of an integrable function.
Best Answer
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.
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.
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