In elementary calculus, a usual argument for the derivative not being defined at an endpoint is that one of the lateral derivatives not being defined. See for example this page 3. Central to the definition of the derivative is that of a limit.
$$\lim_{x\rightarrow a}f(x)=b \Leftrightarrow \forall_{\epsilon}\exists_{\delta}\forall_{x}(x\in D \ \land \ x\in N_{\delta}(a)\ \Rightarrow \ f(x) \in N_{\epsilon}(b) )$$
I fail to see why they say that for an endpoint the derivative is not defined, when clearly using this definition of limit we can speak of continuity at an endpoint... For the left endpoint of a closed interval, the left limit would be defined for any $b$, since the premiss of the implication is false, making the whole implication true for whatever b.
