When $f\in L\_\text{loc}^1$, it's distributional derivative can be defined as $D\_{f'}\in\mathfrak{D}'$, such that $D\_{f'}(\varphi)=-\int f\varphi'$ for all $\varphi\in\mathfrak{D}$, where $\mathfrak{D}$ is the space of test functions. Then from what I understood, $f$ is said **weakly differentiable**, if there exists $f'\in L\_\text{loc}^1$ such that $D\_{f'}(\varphi)=\int f'\varphi$ for all $\varphi\in\mathfrak{D}$. It is also well-known fact that not all locally integrable functions are weakly differentiable, because $L\_\text{loc}^1$ is proper subset (in sense of isomorphism) of $\mathfrak{D}'$. 


I tried to picture these definitions into my head in visual ways (which is not necessary), but having hard time to fully characterize the definition of weak differentiability. What I can say is $f$ must not make any jumps. I also tried to understand the space on the space of distributions : if we let $D(L\_\text{loc}^1) \subset \mathfrak{D}'$ as the set of all distributions such that there exists it's representation $f\in L\_\text{loc}^1$, and $d(L\_\text{loc}^1)\subset\mathfrak{D}'$ a set of all distributional derivatives, then the space of weakly differentiable distributions (which means it's representation in $L\_\text{loc}^1$ is weakly differentiable) will be $W(L\_\text{loc}^1):=D(L\_\text{loc}^1)\cap d(L\_\text{loc}^1)$. But I don't have any idea to characterize this space in other ways. 


Here are the questions : For any given $f\in L\_\text{loc}^1$,


1. Is there any sufficient condition (which is not too trivial, for
example, $f$ is differentiable in classical sense) for $f$ to be
weakly differentiable?
2. Is there any necessary condition (again, not trivial one) for $f$ to be weakly differentiable?
3. In other words, is there any nontrivial characterization of $W(L\_\text{loc}^1)$?


Thank you in advance.



Is there any nontrivial characterization of weakly differentiable functions?
**Definition.**
If $U\subset\mathbb{R}$ is open, we say that $u\in {AC}(U)$
if $u$ is absolutely continuous on every compact interval in
$U$. Let $\Omega\subset\mathbb{R}^n$. We say that
$u$ is *absolutely continuous on lines*, $u\in {ACL}(\Omega)$,
if the function $u$ is Borel measurable and for almost every line
$\ell$ parallel to one of the coordinate axes, $u|\_\ell\in AC(\Omega\cap\ell)$.
Since absolutely continuous functions in dimension one are differentiable a.e.,
$u\in {ACL}(\Omega)$ has partial derivatives a.e.



> 
> **Theorem.** $f\in L^1\_{\rm loc}(\Omega)$ has weak derivative $\nabla f\in L^1\_{\rm loc}(\Omega)$ if and only if $f\in ACL(\Omega)$. Moreover the pointwise derivative of $f$ which exists a.e., equals to the weak derivative.
> 
> 
> 


**Remark.** Functions that are equal a.e. are identified so by writing $f\in ACL(\Omega)$ we mean that $f$ equals a.e. to a function that belongs to $ACL(\Omega)$.


The above result is Theorem 4.21 in [EG] or Theorem 2.23 in [H] or Theorem 1 p. 4 and Theorem 2 p. 6 in [M].


**[EG] L. C. Evans, R. F. Gariepy,** *Measure theory and fine properties of functions.* Revised edition.
Textbooks in Mathematics. CRC Press, Boca Raton, FL, 2015.


**[H]** <http://www.pitt.edu/~hajlasz/Notatki/Cortona%20Lectures.pdf>


**[M] V. Maz'ya,** *Sobolev spaces with applications to elliptic partial differential equations.* Second, revised and augmented edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. 


