Finding an unbounded set with a specific property

Question

Find an unbounded subset $A ⊂ \Bbb R$ such that every function from $A$ to a metric space is uniformly continuous.

My attempt at the solution (incomplete).

If $A⊆ \Bbb R$ were such a set, then for each $x_{0} ∈A$ the function $f_{x_{0}}:A→\Bbb R$ defined by $f_{x_{0}}(x)=\{1$,if $x=x_{0}$

        $0,if $ x≠x_{0} $

is continuous.

I don't know how to proceed further. Please suggest.

Answer 1

Hint:

1. Suppose A has discrete metric, then every function frome it to a metric space is uniformly continuous.

2. Now you are asked to find a set $A\subset \mathbb{R}$, such that every function from to metric space $Y$ it is uniformly continuous in terms of the subspace topology (This is what $f:A\to Y$ is uniformly continuous means). Note the induced subspace topology on a subset of $\mathbb{R}$ can have discrete metric even if $\mathbb{R}$ has standard metric. Now can you find an unbounded subset of $\mathbb{R}$ whose subspace topology is discrete metric?

   $\mathbb{N}\subset \mathbb{R}$ has discrete metric.