Let $\mathbb{X},\mathbb{Y}$ and $T$ metric spaces. A family $\{f_t\}_{t\in T}$ of maps $f_t:\mathbb{X}\to\mathbb{Y}$ is uniformly continuous with respect to parameter $t$ if, $$ (\forall \epsilon>0)(\exists \delta(\epsilon)>0)(\forall x,y\in\mathbb{X}\wedge \forall t\in T)\big( d_\mathbb{X}(x,y)<\delta(\epsilon)\implies d_\mathbb{Y}(f_t(x),f_t(y))<\epsilon \big) $$ A well-known sufficient condition is that $\{f_t\}_{t\in T}$, with $T$ a compact metric space, is uniformly continuous with respect to parameter $t$ iff,
$$ (\forall \epsilon>0)(\exists \delta(\epsilon)>0)(\forall x,y\in\mathbb{X})\big( d_\mathbb{X}(x,y)<\delta(\epsilon)\implies \sup_{t\in T}d_\mathbb{Y}(f_t(x),f_t(y))<\epsilon\big) $$

Another sufficient condition (used in the in the assumptions of Picard's theorem to ensure existence and uniqueness of solutions to ordinary differential equations) is that the function is uniformly Lipschitz with respect to the parameter $t$, i.e. $$ (\ast)\quad(d_{\mathbb{Y}}(f_t(x),f_t(y))\leq M d_{\mathbb{X}}(x,y),\quad \forall x,y\in\mathbb{X},\forall t\in T $$

My question. I wonder if there are alternative criteria ( or sufficient condition ) to tell whether a family $ \{f_t\}_{t \in T}$ of functions $f_t:\mathbb{X}\to\mathbb{Y}$, with $ t\in T $, $ \mathbb{X}$ and $\mathbb {Y} $ metric spaces, has the uniform continuity property with respect to the parameter $t$. For alternative criteria I want to say situations where $ T $ is not compact or that the condition $(\ast)$ is not met.


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