English
In a one-dimensional setting, two line derivatives along the same direction at the same point coincide; hence their difference is zero and they are not independent unless they are equal.
Русский
В однородном случае два линейных производных по одному и тому же направлению совпадают; их разность ноль, они не независимы, если не равны.
LaTeX
$$$$\text{If } HasLineDerivAt 𝕜 f f'_0 x v \text{ and } HasLineDerivAt 𝕜 f f'_1 x v, \; f'_0 = f'_1.$$$$
Lean4
/-- `f` is line-differentiable at the point `x` in the direction `v` if there
exists `f'` such that `f (x + t v) = f x + t • f' + o (t)` when `t` tends to `0`. -/
def LineDifferentiableAt (f : E → F) (x : E) (v : E) : Prop :=
DifferentiableAt 𝕜 (fun t ↦ f (x + t • v)) (0 : 𝕜)
