Uniform Continuity

Definition

Consider a function $f:\mathcal{D}\to \mathbb{R}$. The function $f$ is uniformly continuous in the $\mathcal{D}$ if $\forall ϵ>0,\exists \delta >0,\forall x,y\in \mathcal{D},(|x-y|<\delta \Rightarrow |f(x)-f(y)|<ϵ)$

Facts

If a function is uniformly continuous, then the function is continuous

Continuously differentiable $\subset$ Lipschitz continuous $\subset$ $\alpha -$Holder Continuous $\subset$ Uniformly Continuous $\subset$ Continuous Lipschitz continuous Absolute continuous $\subset$ Uniformly Continuous $\subset$ Continuous where $0<0\le 1$

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