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The space $C^0$ denotes the set of continuous and differentiable functions, the space $C^1$ the set of the continuous and differentiable functions which have a continuous and differentiable first derivative and so on.

But it is well known that there are functions which are continuous, but not differentiable. So why there are no spaces for them, I mean a space of the continuous functions, and a space for the functions which have just continuous derivative (not necessarily differentiable too)?

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What do you mean by there's no space for them? $C$ is continuous functions, and your $C^1$ should be differentiable with continuous first derivative? –  user27126 Jan 12 '13 at 19:07
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Who said it wasn't important? –  copper.hat Jan 12 '13 at 19:12
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Indeed, they're absolutely essential to the theory of $C^\ast$-algebras, and more generally to the story of operator-algebraic noncommutative geometry... –  Branimir Ćaćić Jan 12 '13 at 19:30
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Your notation is wrong: $C^0$ usually denotes the continuous functions, i.e., not necessarily differentiable ones! Analogously, $C^1$ are the at least once differentiable functions with continuous first derivative! –  AlexE Dec 5 '13 at 9:21

2 Answers 2

up vote 9 down vote accepted

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.

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Side note: I think you have problems with notation. Usually $C^0$ denotes the space of continuous functions, $C^1$ the space of differentiable functions with continuous derivative and etc.

Answer: In functional analysis when we are talking about spaces we mean linear spaces. The space of continuous nondifferentiable functions is not linear because it doesn't contain, for example, zero function.

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What now, is $C^0$ the space of just continuous functions or of continuous and differentiable functions? (if $C^k$ denotes the continuous functions which are $k$-th order differentiable with the $k$ derivative being continuous than the second option would be denoted by $C^0$, which is a more restrictive set than just the continuous functions...) –  Stefan Jan 13 '13 at 13:07
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@Stefan definition is inductive $$C^0(M)=\{f:M\to\mathbb{C}:f\mbox{ is continuous }\}$$$$C^k(M)=\{f\in C^{k-1}(M):\exists f'\in C^0(M)\}$$ –  Norbert Jan 13 '13 at 13:23
    
And what about the space of functions, that are just differentiable (where the derivative is not necessarily continuous)? –  Stefan Jan 13 '13 at 14:09
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@Stefan: If you're interested, On nowhere monotone functions by Clifford E. Weil [Proc. Amer. Math. Soc. 56 (1976), pp. 388-389] makes use of the space of bounded derivatives (not functions that have a derivative, but rather the derivatives themselves). –  Dave L. Renfro Jan 16 '13 at 20:30
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@Norbert: Your definition of $C^k(M)$ is a bit odd. Wouldn't it be better to write $D(M)$ for the set of differentiable functions on $M$ and then $C^k(M) = \{f \in D(M) : f' \in C^{k-1}(M)\}$? –  kahen Jan 24 '13 at 10:05

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