So i have a set of all real-number sequences called M, and D is the "comparing" metric, since it gives "distance" between two sequences as index of elements on which those two differ. And there is also d is discrete metric on $\mathbb{R}$ and $d_{2}$ is the Euclid metric on $\mathbb{R}$
So i need to investigate continuity,uniform continuity and Lipschitz continuity for functions:
$F:(M,D)\rightarrow(\mathbb{R},d)$
$F:(M,D)\rightarrow(\mathbb{R},d_2)$
Where $F(x)=x_{2016}$ for any $x=(x_1,x_2,x_3,...) \in M$
Since distance between two sequences is: $0<D(x,y)<1$, i should just look at what is happening after function effects those sequences and try to determine wheter it's continuous or even uniform or Lipschitz continuous.
But i am troubled with how should i determine that, since i don't know wheter to start at Lipschitz part or at continuous part, also i can only use some basics regarding all this, mostly by definition and tricks regarding definition of continuity and both of the other more advanced versions of continuity.
Any help, hint or even answer to my question would be appreciated.
Also, would i need to check Lipschitz continuity at first?