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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?

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up vote 1 down vote accepted

Hint: $\lVert (t+rk,s)-(t,s)\rVert\leq |rk|\leq |k|$ if $0\leq r\leq 1$. So apply the definition of uniform continuity to $\varepsilon$ to get a corresponding $\delta$. Then take $|k|\leq \delta$.

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