Math and science::Analysis::Tao::10: Differentiation of functions
Differentiability on a domain
Let \( X \) be a subset of \( \mathbb{R} \) and let \( f : X \to \mathbb{R} \) be a function. We say that \( f \) is differentiable on \( X \) iff for every limit point \( x_0 \in X \), \( f \) is differentiable at \( x_0 \) on \( X \).
Tao describes this as an if rather than iff statement.
I think this definition is allowing isolated points to be ignored. Note how the limit point is also an element, so we are also excluding points that are limit points but not in the \( X \).