Do limits of (continuous) functions have a meaning in the distribution sense?

Question

This question came to my mind when working with distributions. We know that locally integrable functions can be realized as distributions. Particularly, if $f$ is locally integrable on $\Omega$, then for any $\phi \in C^{\infty}_c ( \Omega )$, we have the action of $f$ on $\phi$ given by

$$f ( \phi ) = \int\limits_{\Omega} f ( x ) \phi ( x )\, \mathrm{d}x.$$

Now, we also know that continuous functions are locally integrable, and hence can be realized as distributions. My question is whether we can make sense of the point values $f ( x )$ as distributions? Further, can we make sense of "$\lim\limits_{x \rightarrow x_0} f ( x ) = f ( x_0 )$" in the sense of distributions? Any insights into this are appreciable!

Answer 1

For a distribution $f$, it doesn't make much sense to talk about its value at a given point $x$. The behavior of $f$ can be certainly localized, but still you need "infinitesimally small neighborhood of $x$" in order to properly describe the behavior of $f$ about $x$. In fact, you can even glimpse this issue when $f$ is a locally integrable function. An element of $L_{\text{loc}}^1(\Omega)$ is an equivalence class of functions that is insensitive of modification of function values on null sets. This renders the idea of "the value of $f(x)$" for locally integrable $f$ essentially meaningless.

If the point evaluation of a distribution is not the right choice, what will be the right type of data for encoding local information of a distribution? Also, once this issue has been addressed successfully, how do we define the limit of such data? One may imagine that the notion of germ might help. Unfortunately, however, germ is too restrictive for defining any useful topology on it. This means that we somehow need less sensitive way of localizing distributions. To this direction, googling seems to suggest that the following definition is a useful choice:

Definition. Let $f$ and $g$ be distributions on $\mathbb{R}^d$. Then we say that $f \sim_x g$ if

$$ \lim_{\lambda \to 0^+} |f(\varphi_x^{\lambda}) - g(\varphi_x^{\lambda})| = 0 $$

for any $\varphi \in \mathcal{D}(\mathbb{R}^d)$, where $\varphi_x^{\lambda}(z) = \lambda^{-d}\varphi(\lambda^{-1}(z - x))$.

(Remark. A uniform version of this definition has been used to formulate and prove Hairer's reconstruction theorem for distributions.)

Let us make several observations:

1. Clearly $\sim_x$ is an equivalence relation, which depends only on the restriction of $f$ and $g$ on arbitrary neighborhood of $x$. In other words, equivalence classes of $\sim_x$ is determined solely on the germs of distributions at $x$. We will denote the equivalence class of $f$ for $\sim_x$ by $[f]_x$.

2. If $f$ is a locally integrable function such that $\lim_{y \to x} f(y) = \ell$, then $[f]_x = [\ell]_x$ (where $\ell$ in RHS now represents the constant function with value $\ell$ everywhere). So, the relation $\sim_x$ captures the notion of pointwise value of continuous function.

3. Let $\mathcal{D}'_x$ denote the collection of all equivalence classes for $\sim_x$. Then we equip $\mathcal{D}'_x$ with the topology generated by the pseudometrics of the form $$ d_{\varphi}([f]_x, [g]_x) := \min\Bigl\{ 1, \, \limsup_{\lambda \to 0^+} |f(\varphi_x^{\lambda}) - g(\varphi_x^{\lambda})| \Bigr\} $$ for $\varphi \in \mathcal{D}(\mathbb{R}^d)$. It is routine to prove that $d_{\varphi}$ is indeed a well-defined pseudometric on $\mathcal{D}'_x$. Moreover, the resulting topology on $\mathcal{D}'_x$ is Hausdorff.

4. For each $x \in \mathbb{R}^d$, we introduce the translation operator $T_x$ on distributions defined by $$T_x f(\varphi(\cdot)) = f(\varphi(\cdot + x))$$ and then identify $[f]_x$ with $[T_{-x}f]_0$. This makes sense, because if $f$ is a locally integrable function such that $\lim_{y \to x} f(y) = \ell$, then $[T_{-x}f]_0 = [\ell]_0$.

Using this formulation, we can prove:

Theorem. Let $f \in C(\mathbb{R}^d)$. Then $$ \lim_{y \to x} \, [f]_y = [f]_x. $$

Proof. For any $\varphi \in \mathcal{D}(\mathbb{R}^d)$, it follows that

$$ d_{\varphi}([f]_y, [f]_x) = d_{\varphi}([T_{-y}f]_0, [T_{-x}f]_0) = \min\left\{ 1, \, |f(y) - f(x)| \left| \int_{\mathbb{R}^d} \varphi(z) \, \mathrm{d}z \right| \right\} \to 0 $$

as $y \to x$. Therefore the desired claim follows. $\square$