A function defined for all inputs? This might seem like a weird question, but is it actually possible to define a function for all possible inputs? By this, I really mean /all/ possible inputs, including numbers, true and false, sets, sets of sets, other functions, itself---everything. To me, this doesn't seem problematic, but maybe there's some kind of subtle reason why this cannot be done. Here's an example (maybe) of a function defined for all possible inputs:
$F(x) = \mathbf{true} \text{, if } x = 0 \\
F(x) = \mathbf{true} \text{, if } x = 1 \\
F(x) = \mathbf{false} \text{, otherwise.}$
Is there anything wrong with that?
 A: A function is normally understood to have domain that is a set of some kind.  Doing so allows us to handle functions as sets themselves, namely sets of ordered pairs.  For example, $f(x)=x^2$ on the integers can be thought of as $\{(1,1), (2,4), (-2,4),\ldots\}$.
However not all mathematical objects are sets; for a famous example consider Cantor's paradox, i.e. if you take all sets, the result is not a set.  This object can be called a class, and such objects are so large and weird that we can't really define functions on them in the usual way.  We can define function-like things, as done in the OP.  However if we allow the domain to not be a set, then the resulting function is not a set.  Thus it lives outside of familiar set theory.  This makes it very unusual, since (apart from logicians) most people live their entire mathematical lives using objects that are defined within a set theory, typically ZFC or something similar.
A: Your $F$ is actually a logical predicate:
$\forall x:[F(x)\iff x=0\lor x=1]$
In mathematics (if maybe not in philosophy), you would probably want to restrict the domain of quantification (e.g. to the set of natural numbers $\mathbb{N}$) as follows:
$\forall x\in \mathbb{N}:[F(x)\iff x=0\lor x=1]$
A: Your approach is fine as long as you know how $0$ and $1$ are represented.  An even simpler approach is $F(x)=2$ where you ignore the input entirely.  All that you need for a function is a unique response to every input in the domain.  Why pay attention to the input?
