The Derivative of a Vector-Valued Function as a Limit of Difference Quotients in Norm

In the setting of Multivariable Calculus: Notation, let $q\in\mathbb{N}$, let $I$ be an interval containing two distinct points, as in Single-Variable Calculus on an Interval (ii), let $\gamma:I\to\mathbb{R}^{q}$, let $t\in I$ and let $c\in\mathbb{R}^{q}$.

(i) Difference quotients in norm. $\gamma$ is differentiable at $t$ with $\gamma'(t)=c$ if and only if for every real $\varepsilon>0$ there is a real $\delta>0$ such that every $s\in I$ with $0<|s-t|<\delta$ satisfies

$$\Bigl\lVert\frac{\gamma(s)-\gamma(t)}{s-t}-c\Bigr\rVert<\varepsilon .$$