Differentiability
Derivative at a point: Let I denote the open interval (a, b) in R, and let x₀ ∈ I. Then a function f: I → R is said to be differentiable at x0, if
exist (finitely) and is denoted by f′(x0).
Differentiability in an Open Interval
A function f is said to be differentiable in an open interval (a, b), if it is differentiable at each point of the open interval.
Differentiability in a Closed Interval
A function f : [a, b] → R is said to be differentiable in closed interval [a, b] if it is
1. differentiable from right a t a [i.e. R f′(a) exists] and
2. differentiable from left at b [i.e. L f′(a) exists] and
3. differentiable in the open interval (a, b).
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