A differentiable function of one real variable is a function whose derivative exists at each point in its domain. Would the derivative still exists if the domain of the function is the null set?
For example, the function
$$ f(x)= \left( \sin \frac{1}{x} \right)^{\cos x} + \arcsin (4^x) + \frac{\sin x}{\ln x} $$ is undefined for any real values of $x$, hence, its domain is $\varnothing$. Are functions like this still differentiable at this condition?
This is really an answer to the comment, but I'll post this as an answer because it is too long for a comment and probably also resolves the question in the OP.
The empty function is continuous by definition. Indeed, according to your definition every $c \in \varnothing$ satisfies the requirement. That may look a little bit odd, but think about it this way: The function is continuous at no point. So the set of points where the function is continuous is the empty set. As the empty set is all you have, the function is continuous everywhere. It's just one of these oddities with the empty set like "all cows on the moon are green" is a true statement.
Similarly, you can argue that the empty function is differentiable.
