I have to prove that a function $f: M \rightarrow N$ where $(M,d)$ and $(N,p)$ are metric spaces is continuous, given it is Lipschitz continuous. What I found so far:
Let $\epsilon>0$ and $\delta=\epsilon/K$
Since $f$ is Lipschitz continuous, there exists a $K<\infty$ such that $p(f(x),f(y)) \le Kd(x,y) = K\delta < \epsilon$
Hence $p(f(x),f(y)) < \epsilon$ for $d(x,y)<\delta$ and thus $f$ is continuous.
It feels like it is too easy and I forgot something. Can someone help me?