TITLE: Is differentiability defined at an isolated point of the function's domain?
QUESTION [5 upvotes]: Say we have the function $f:(0,1)\cup\{2\}\rightarrow \mathbb R$ defined by $f(x)=5$.
Is $f$ differentiable or not differentiable (or neither) at $2$?
Is $f$ a differentiable function?

REPLY [2 votes]: Differentiability is defined in terms of sequences of differences $h_1,h_2,h_3\dots$ that are all nonzero and yet converge to zero. Such a sequence does not exist in the given domain, as there does not exist a number $k$ in the domain where $|2-k|=0.2$ or $0.1$ or indeed any number smaller than $1$. Hence differentiability is not defined at isolated points.