# Functions in Sobolev Spaces that are NOT continuous

For $$\beta\in N_{+}$$, $$d\in N_{+}$$ define the Sobolev space $$\mathcal{W}^{\beta,\infty}([-1,1]^d)$$ of functions $$f:[-1,1]^d \to \mathbb{R}$$ such that $$f$$ and its weak partial derivatives up to order $$|\beta|$$ are elements of $$L^{\infty}$$-the space of essentially bounded functions.

Sobolev Imbedding Theorems (such as Thrm 4.12 of Adams and Fournier, 2003) seem to indicate that functions in this Sobolev space are continuous, $$\mathcal{W}^{\beta,\infty}([-1,1]^d) \subset C^0([-1,1]^d)$$.

# Question:

I have ran across what seems to be a counterexample to this result.

Let $$\mathbb{Q}$$ denote the rational numbers, and consider the function $$\boldsymbol{1}_{\mathbb{Q}}(x)$$ which is equal to one if $$x\in \mathbb{Q}\cap [-1,1]$$ and is equal to zero if $$x\in [-1,1] \setminus \mathbb{Q}$$. Clearly this function is bounded so it is in $$L^{\infty}$$ and it is well known that this function has a weak derivative $$v(x)=0$$.

It follows that this function should be an element of $$\mathcal{W}^{1,\infty}([-1,1])$$, however it is not continuous which seems to be a violation of $$\mathcal{W}^{\beta,\infty}([-1,1]^d) \subset C^0([-1,1]^d)$$.

Can someone explain to me what is going on here? Is this a counterexample or is one of my conclusions wrong?

(Additional references on this topic would also be much appreciated)

That is a fine technicality of $$L^P$$ ($$p=\infty$$ included) spaces. Technically, you are using equivalence classes of functions: $$f \sim g \iff f=g \; \textrm{a.e.}$$ In your case $$0 \sim 1_\mathbb{Q}$$ and the Sobolev embedding theorem tells you that there is a continuous representative in this class of functions. In your case, the $$0$$ function.

From that perspective, you could take any $$f \in C_c^{\infty}([0,1])$$ and add $$f+1_\mathbb{Q}$$ to make it nowhere continuous, even though all your Sobolev embeddings apply.

It is therefore common to always refer to the "most regular representative" in an equivalence class of $$L^p$$-functions. See the comments to see what this means precisely.

• Nice! This doubt is common. The immersion $W^{k,p} \hookrightarrow L^p$ means that for all $u \in W^{k,p}$ there exists a function $u^* \in L^p$ such that $u=u^*$ a. e. Sep 13, 2022 at 22:30
• Note that the most regular representative of an element of $L^1_{loc}$ is equal to $\lim_{\epsilon \to 0^+} \frac{1}{m(B_\epsilon(x))} \int_{B_\epsilon(x)} f(y) dm(y)$ whenever this limit exists. It doesn't really matter all that much what it is where this limit doesn't exist.
– Ian
Sep 13, 2022 at 22:31
• Yes, both of your comments give a nice definition what I called "most regular representative". And I had this doubt too when I was first introduced to Sobolev spaces. Sep 13, 2022 at 22:34
• When writing the most regular representative as @Ian did does this mean for $f\in L^{1}_{loc}$ the most regular representative of $f$ is $g$ such that $g(x) = \lim_{\epsilon\to 0^{+}} \frac{1}{m(B_{\epsilon}(x))} \int_{B_{\epsilon}(x)}f(y)dm(y)$? Sep 13, 2022 at 23:02
• @ChadBrown That is what I meant, yes.
– Ian
Sep 13, 2022 at 23:46