> [!Definition] > We say that a given function $f: \mathbb{R} \to \mathbb{R}$ is *a Lipschitz Function* or is *Lipschitz continuous* with *Lipschitz constant* $L>0$ if $\forall x,y \in \mathbb{R}, \quad |f(x)-f(y)| \leq L|x-y|.$ > # Properties > [!info] > Any function with a bounded first derivative must be Lipschitz. See [[Uniqueness Theorem for Explicit First Order Initial Value Problem]] > > [!Lemma] > Lipschitz Continuous $\implies$[[Continuous Real Function|Continuous]] on $\mathbb{R}$. > *Proof*. Take $x,c \in \mathbb{R}$ and $\varepsilon>0.$ Whenever $|x-c|<\delta := \frac{\epsilon}{L} \implies |f(x) -f(c)| \leq L|x-c| < \epsilon$ > Since $\delta>0$, this shows that $f$ is continuous at every $c \in \mathbb{R}$.