Suppose $f_t$ (and $f$) is uniformly continuous on $[0,T] \times S$, where $S \subset \mathbb{R}^n$ is compact.
The uniform continuity allows us to estimate $\lVert f(t+k, \cdot) - f(t, \cdot) - f_t(t, \cdot)k \rVert_\infty = \lVert k \int_0^1 f_t(t+rk, \cdot) - f_t(t,\cdot)\;dr\rVert_\infty \leq k\epsilon$ for $k$ small enough.
I'm a bit stumped. The second equality I can see by just integrating it out. But why the estimate hold? What's the intuition?