Uniformly continuous function Let $f:[0,1] \cup \{-1\} \to \Bbb R$ be defined by, 
$f(x)=1 , \forall x \in [0,1]$;
$=0$ ,for 
$x=-1$.
Is the function uniformly continuous? 


*

*here the domain is closed and bounded. Thus compact. And also f(x) is continuous.  Thus f should be uniformly continuous. But here how the definition of uniform continuity is applicable? I can't understand it. Or is it uniformly continuous at all?


The "same" reason you saw it is continuous. The function is uniformly continuous on $[0,1]$ clearly. If $x,y \in \{ -1 \}$, then $|x-y| = 0$ and $|f(x)-f(y)| = 0$. So $f|_{\{-1\}}$ is such that for every $\varepsilon > 0$ and every $\delta > 0$ we have $|x-y| < \delta$ implying $|f(x)-f(y)| < \varepsilon$; this is even stronger than the requirement of uniform continuity.
