Differentiability at a point, definition
Differentiability at a point
Let \( X \) be a subset of \( \mathbb{R} \), and let \( x_0 \in X \) be an element of \( X \) that is also a limit point of \( X \). Let \( f : X \to \mathbb{R} \) be a function. Then, if the following limit:
converges to some \( L \) then we say that \( f \) is differentiable at \( x_0 \) on \( X \) with derivative \( L \). We write: \( f'(x_0) := L \).
If \( x_0 \) is not an element of \( X \) or is not a limit point of \( X \), or the limit does not converge, we leave \( f'(x_0) \) undefined and we say that \( f \) is not differentiable at \( x_0 \) on \( X \).
We need \( x_0 \) to be a limit point (not isolated) in order for it to be adherent to \( X \setminus \{x_0\} \) and in turn for the above limit to be defined. Thus, functions do not have a derivative defined at isolated points.