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?