Personally I would never write "
", but if someone does write that (on an exam solution or something) it is just a sloppy way of saying "the function 
does not approach a limit as 
approaches 
."
Like, if you were to write
Ln(-1) =
Personally I would never write "", but if someone does write that (on an exam solution or something) it is just a sloppy way of saying "the function does not approach a limit as approaches ."
restatement of, and direct answer to, the Question:
Let $c:= \mathtt{«}\text{an expression with }x\mathtt{»} \quad$and$\quad \mathtt{DNE}:=\mathtt{«}\text{an arbitrary nonexistent object}\mathtt{»}$.
Then, from that, the answer to
Can
?
is, technically, Yes, iff you make adequate logical justifiable assumptions (but No if you faultily allow some loose logical construction, or provide no reasoning at all), for the cases when 
does not exist. Sufficient logical justification is possible but lengthy (and moreover most likely skipped and hence not recognized or simply considered notationally wrong, since nonstandard), whereas an alternative notation is much more automatically understandably correct.
Logical justification of the affirmative:
Let $\mathtt{«}\text{an arbitrary nonexistent object}\mathtt{»}:=d$. Then by transitivity, 
.
But how is 
defined?
By definition, $∄d$. In a manner of speaking, this
$\quad ⟹\quad d∈\{\}\:⇔\:∅∋d\quad \quad $which seems to contradict the literal construct defining the nullset as empty i.e. having no members. This could be justified by recognizing as, and proceeding to define to be, 
as ‘null_element’; parallel to how $∅$ is defined to be the ‘null_set’.
In light of this sort of leap in logic, I would propose a different symbol for '
' as used herein: the lowercase slashed-o (
), since the uppercase version (
) looks typographically similar to the emptyset symbol.
Of course, this still leaves a gap in converting “DNE” to “ø”, where ‘DNE’ is taken to be an object of some sort rather than a qualitative state such as a group action.
further considerations, with possibly better formal notation:
Read aloud in non-abbreviated form, "DNE" as used here (a limit that doesn't converge ergo doesn't exist) is simply its de-initialism “Does Not Exist”, rather than “Non-Existent Entity” or the like. As such, "
" ought to be read as “c equals does-not-exist” or “c is equal to does_not_exist”, which is ungrammatical. On the otherhand, one could simply slap the nonexistential quantifier in place of the equals sign & RHS, and call it a day: "
". Problem with this is that placing quantifiers after instead of before the modified phrase is non-standard, especially in a standalone expression not contingent on something else. Thus instead "$∄c$" would be more notationally correct, despite its following a different “grammatical form” than in the existent-type case (since if $∃c$, then unless you don't yet know the existent value then you would more likely want to indicate a less-vacuous $c=\mathtt{«}\text{its value[s]}\mathtt{»}$). Considering this, it would make more linguistically and notationally valid sense to replace ‘
’ with ‘
’ (omitting, or for typographic clarity instead substituting with a space or two, the ‘
’).
If you would like to retain the general flow of the mathematical statement in the does-not-exist as in the does-exist[-and-equals-«blah»] case, but think that following the expression by the freestanding letters “DNE” is aesthetically displeasing, then you could probably be able to convey the meaning (in consistent form) by swapping the equality-symbol with some other relational symbol and an appropriate glyph or glyphs to the right of that. I’m not sure what the most technically correct & appropriate option would be (and would appreciate Comments on the matter), but perhaps an equivalence relation (‘≡’) or strong_entailed-by(‘⫤’) could do the trick, such as “$c≡\mathtt{DNE}$” or perhaps “$c⫤∅$”.
According to a Wikipedia article on the subject, in Herbert B. Enderton's book Computability: An Introduction to Recursion Theory (2011), even if nowhere else (no other reference is given, and I've never seen the usage):
If
is a partial function on
and
is an element of
, then this is written as
and is read as "
is defined."
If
is not in the domain of
, then this is written as
and is read as "
is undefined".
I have never seen such a symbol. I don't think it would be very useful, and it might make unexperienced people less aware that they are dealing with an undefined entity, and start doing calculations with it getting meaningless results.
Equal sign with vertical line
The vertical line | is a little tall for my taste. The following definition for \vneq decreases the total height of the vertical line to match the total height of \neq. Resizing vertical height will not change the line thickness in horizontal direction.
- The final witdh and height of the vertical line can be fine-tuned by redefining macros
\vneqxscaleand\vneqyscale. The default is1. \mathpaletteallows the symbol to resize automatically.
Example file:
\documentclass{article}
\usepackage{amssymb}% \varnothing
\usepackage{graphicx}% \resizebox
\makeatletter
\newcommand*{\vneq}{%
\mathrel{%
\mathpalette\@vneq{=}%
}%
}
\newcommand*{\@vneq}[2]{%
% #1: math style (\displaystyle, \textstyle, ...)
% #2: symbol (=, ...)
\sbox0{\raisebox{\depth}{$#1\neq$}}%
\sbox2{\raisebox{\depth}{$#1|\m@th$}}%
\ifdim\ht2>\ht0 %
\sbox2{\resizebox{\vneqxscale\width}{\vneqyscale\ht0}{\unhbox2}}%
\fi
\sbox2{$\m@th#1\vcenter{\copy2}$}%
\ooalign{%
\hfil\phantom{\copy2}\hfil\cr
\hfil$#1#2\m@th$\hfil\cr
\hfil\copy2\hfil\cr
}%
}
\newcommand*{\vneqxscale}{1}
\newcommand*{\vneqyscale}{1}
\makeatother
\begin{document}
\[
% Comparison \neq vs. vneq
\varnothing \neq \emptyset \vneq \varnothing \\
\]
\[
% Check sizes:
\vneq^{\vneq^{\vneq}} \\
\]
\[
% Bounding box checks:
\setlength{\fboxsep}{0pt}
\setlength{\fboxrule}{.1pt}
\fbox{$\neq$}\,\fbox{$\vneq$}\,\fbox{$|$}
\]
\end{document}
The height can be further decreased, e.g.
\renewcommand*{\vneqyscale}{.8}
Result for mathabx:
Result for txfonts:
Result for MnSymbol:
Here the vertical line is too thick and the horizontal resizing needs shrinking:
\renewcommand*{\vneqxscale}{.67}
Result for MnSymbol and \vneqxscale = .67:
Alternative to varnothing
Instead of changing \neq, the empty set symbol \varnothing could be constructed using \not to match the slope of the slanted vertical lines.
However, \circ is too small and \bigcirctoo big. Therefore this method is shown for txfonts that provides \medcirc and MnSymbol with \medcircle.
\documentclass{article}
%\usepackage{txfonts}
%\newcommand*{\varemptysetcircle}{\medcirc}
\usepackage{MnSymbol}
\newcommand*{\varemptysetcircle}{\medcircle}
\makeatletter
\newcommand*{\varemptyset}{%
{% mathord
\vphantom{\not=}% correct height and depth of the final symbol
\mathpalette\@varemptyset\varemptysetcircle
}%
}
\newcommand*{\@varemptyset}[2]{%
% #1: math style (\displaystyle, \textstyle, ...)
% #2: circle
\ooalign{%
\hfil$\m@th#1\not\hphantomeq$\hfil\cr
\hfil$\m@th#1#2$\hfil\cr
}%
}
% \not can be redefined to take an argument
\newcommand*{\hphantomeq}{%
\mathrel{\hphantom{=}}%
}
\makeatother
\usepackage{color}
\begin{document}
\[
\not=\; \color{blue}\neq \varemptyset\; \color{black}\varnothing
\]
\end{document}
Result for txfonts:
Result for MnSymbol:
Yes:
\documentclass[a5paper]{article}
\usepackage{amssymb}
\usepackage{amsmath}
\newcommand\vneq{\mathrel{\ooalign{$=$\cr\hidewidth$|$\hidewidth\cr}}}
\begin{document}
\begin{align*}
a&\gvertneqq b\\
C&\neq \varnothing \\
d&\vneq f
\end{align*}
\end{document}
For a motivation behind the commands in \vneq, read egreg's excellent tutorial on \ooalign in \subseteq + \circ as a single symbol (“open subset”)