u_t is continuous in t

Proposition

• $y:[t_0-\tau ,t_0+\delta ]\to {\mathbb{R}}^N$: cont. (and hence unif. cont.)
• $\mathrm{\Phi }(t,y)\mapsto y_t(\theta ):=y(\theta +t)$: defined on $[t_0,t_0+\delta ]$

Then $\mathrm{\Phi }$ is cont. in $t$.

Proof

Since $y$ is unif. cont. on the closed interval. $◼$
(c.f. Note)
(c.f. NTHU MATH 526500 DDE Note week 3)