Strict differentiability

Definition

Let $U \subseteq \mathbb{R}^n$ be open, let $f \colon U \to \mathbb{R}^m$, and let $a \in U$. The function $f$ is strictly differentiable at $a$ if there is a linear map $L \colon \mathbb{R}^n \to \mathbb{R}^m$ such that $$\lim_{\substack{(x, y) \to (a, a) \\ x \ne y}} \frac{\|f(x) - f(y) - L(x - y)\|}{\|x - y\|} = 0.$$

Remarks

If $f$ is strictly differentiable at $a$, put $y = a$: then $f$ is differentiable at $a$ and $L = Df(a)$ is its Fréchet derivative. The converse is false. The function $f(x) = x^2 \sin(1/x)$ for $x \ne 0$, $f(0) = 0$, is differentiable at $0$, but it is not strictly differentiable at $0$ because $f'$ is not continuous at $0$. A function of class $C^1$ on $U$ is strictly differentiable at each point of $U$. The same definition works for maps between Banach spaces, with norms in place of Euclidean norms.

Explanations

L1 human

Ordinary differentiability compares $f(x)$ with $f(a)$ only. Strict differentiability compares $f(x)$ with $f(y)$ for all pairs of points near $a$. So the linear approximation is uniform in a small neighbourhood, not only at the single point.