Define $f:[a.b] \rightarrow \mathbb{R}$ as Lipschitz
By definition, then, $\exists M>0, M \in \mathbb{R}:|f(x)-f(y)\leq M|x-y|$ $ $ $\forall x,y\in[a,b]$
Rearranging this, we have $\frac{|f(x)-f(y)|}{x-y} \leq M$, positing that the slope must be finite on the interval $[a,b]$
I understand conceptually that Lipschitz functions must be continuous but I'm having trouble showing it.
Should I go about this by showing assuming the negation of Lipschitz and showing that for some $\epsilon, \exists \delta>0: |f(x)-L|>\epsilon$ ?
Best Answer
It is not necessary to proceed by contradiction. Use the $\varepsilon-\delta$ definition of continuity and choose $\delta=\varepsilon/M$.