Let $\omega\in D'(\mathbb R^n)$ be a distribution and $p\in \mathbb R^n$. If there is an open set $U\subset \mathbb R^n$ containing $p$ such that $\omega|\_U$ is given by a continuous function $f\in C(U)$, then for every $\phi\in C^\infty\_c(\mathbb R^n)$ with $\int\_{\mathbb R^n}\phi(x)d x=1$ we can define a Dirac sequence $\{\phi^p\_j\}\_{j\in \mathbb N}\subset D(\mathbb R^n)$ by $\phi^p\_j(x):=j^n\phi(j(x-p))$ which fulfills
$$
\omega(\phi^p\_j)\to f(p)\quad \text{ as }j\to \infty.
$$
This shows that we can recover the value $\omega(p)\equiv f(p)$ of the distribution $\omega$ at the point $p$ via a limit of such Dirac sequences.


Now, suppose that for some $\omega\in D'(\mathbb R^n)$ and $p\in \mathbb R^n$ we just know that
$
\lim\_{j\to \infty}\omega(\phi^p\_j)
$ 
exists for every $\phi\in C^\infty\_c(\mathbb R^n)$ with $\int\_{\mathbb R^n}\phi(x)d x=1$ and is independent of $\phi$. In view of the above it then seems reasonable to define $\omega(p):=\lim\_{j\to \infty}\omega(\phi^p\_j)$ and to say that $\omega$ has a well-defined value at the point $p$.


**Q:** Is this definition useful in any sense? I have the feeling that it might be fundamentally flawed. In that case, I'd find it interesting to know what's the greatest generality in which one can make sense of "the value of a distribution at a point".


**Additional thoughts after 1st edit:** Some "consistency checks" for the definition would in my opinion be the following: 


1. If the value of $\omega$ exists at every point in some open set $U\subset \mathbb R^n$ and the function $f$ defined on $U$ by these values is continuous, then $\omega|\_U$ is given by $f$.
2. If the value of $\omega$ exists at Lebesgue-almost every point in some open set $U\subset \mathbb R^n$ and the values define a function $f\in L^1\_{\mathrm{loc}}(U)$, then $\omega|\_U$ is given by $f$.


I believe that at least property 1 should be true and I'll check it once I find the time.


**2nd edit**: My question is related to [this MO question](https://mathoverflow.net/questions/262606/distribution-that-vanishes-against-approximated-delta-is-zero?rq=1) which corresponds to the case $f\equiv 0$.



Defining the value of a distribution at a point
The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $\delta$-sequences, follows the path traced years ago by Stanisław Łojasiewicz in the paper [1], so I describe his approach to the problem below.


Łojasiewicz analyzes the problem for functions of one variable, i.e. $n=1$: by using the definition of *change of variables in a distribution* (see for example [2], §1.9 pp. 21-22) and considering the change of variable $y=x\_0+\lambda x$, for $ x,x\_0,\lambda \in\Bbb R$, i.e.
$$
\begin{split}
T(x\_0+\lambda x)&\triangleq \langle T(x\_0+\lambda x),\varphi(x)\rangle\\
&=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x\_0)\big)}{\lambda}\right\rangle
\end{split}
\quad \varphi\in\mathscr{D}(\Bbb R)
$$
he defines the *limit of a distribution at a point $x\_o$* as ([1], §1 p. 2-3)
$$
\lim\_{x\to x\_0} T\triangleq \lim\_{\lambda\to 0} T(x\_0+\lambda x)
\label{1}\tag{1}
$$
and proves that


* $\lim\_{x\to x\_0} T=\lim\_{x\to x\_0^+} T=\lim\_{x\to x\_0^-} T$
* by using an earlier result of Zieleźny, if the limit \eqref{1} exists, it is necessarily a constant $C\in \Bbb C$, or more precisely a *constant distribution* $C$.
* a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $T=f^{(n)}$, where $f\in C^0(\Bbb R)$ and
$$
\lim\_{x\to x\_0}\frac{f(x)}{(x-x\_0)^n}=\frac{C}{n!}.
$$
Then ***Łojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point***: note that this definition does not rely on any particular test function (or sequence of such) $\varphi\in\mathscr{D}(\Bbb R)$, as stated above. Now a few observations:


1. Łojasiewicz ([1], §1 p. 1) states that the case $n>1$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) *Multivariable Analysis*, Springer-Verlag.
2. Łojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7).
3. The term $\lambda^{-1}$, more or less intrinsically used in \eqref{1}, suggests the possible use of the [Mellin transform](https://en.wikipedia.org/wiki/Mellin_transform): this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines the *(generalized) spectral value of a function/distribution at a point* and proves ([3], §12 p. 43) that it coincides with Łojasiewicz point value \eqref{1} when the latter exists (by using the necessary and sufficient condition above): however, the construction of Ziemian does not apply to all distributions.


**References**


[1] Stanisław Łojasiewicz (1957-1958), "[Sur la valeur et la limite d'une distribution en un point](http://matwbn.icm.edu.pl/ksiazki/sm/sm16/sm1611.pdf)" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, [MR0087905](http://www.ams.org/mathscinet-getitem?mr=MR0087905) [Zbl 0086.09405](https://zbmath.org/?q=an%3A0086.09405).


[2] V. S. Vladimirov (2002), *[Methods of the theory of generalized functions](https://books.google.it/books?id=6UpZDwAAQBAJ&printsec=frontcover&hl=it)*, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, [MR2012831](http://www.ams.org/mathscinet-getitem?mr=MR2012831), [Zbl 1078.46029](https://zbmath.org/?q=an%3A1078.46029).


[3] Bogdan Ziemian (1988), "[Taylor formula for distributions](http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-12bb5d6a-bf89-45ae-82f6-d4843817119b/c/rm26401.pdf)", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, [MR0931848](http://www.ams.org/mathscinet-getitem?mr=MR0931848), [Zbl 0685.46025](https://zbmath.org/?q=an%3A0685.46025).


