real analysis – what is the difference between Strong Frechet derivative and Frechet derivative

I am confused about the relationship between Frechet differentiable and strong Frechet differentiable.

Assume the function $f(x) in mathbb{R}, xin mathbb{R}^n$ is Frechet differentiable at $x$.

Then we have $$lim_{x neq y^1 to x} frac{f(y^1) – f(x) – nabla f(x)^T(y^1-x)}{Vert y^1-xVert} = 0$$

$$lim_{x neq y^2 to x} frac{f(y^2) – f(x) – nabla f(x)^T(y^2-x)}{Vert y^2-xVert} = 0$$

Then use the first equation minus the second one and the inequality $Vert y^1-xVert leq Vert y^1-y^2Vert + Vert y^2 – xVert$, then we get

$$lim_{substack{y^1 neq y^2 \ (y^1, y^2) to (x, x)}} frac{f(y^1) – f(y^2) – nabla f(x)^T(y^1-y^2)}{Vert y^1 – y^2 Vert} = 0, $$
which is the definition of strong F-derivative.


It seems I did not use any other condition but F-derivative to get strong F-derivative. I am quite confused about their difference.