I´m reading some notes about differential topology. In some point it says the following: if $Y\subset\mathbb{R}^N$ is a embedded submanifold and $U$ is a tubular neighborhood of $Y$, then there exist a positive function $\epsilon:Y\to\mathbb{R}$ such that $U_\epsilon=\{w\in\mathbb{R}^N\;|\;|w-y|<\epsilon(y)\;\text{for some $y\in Y$}\}$ is contained in $U$.
To prove this assertion, it defines $\epsilon(y)=\sup\{r\;|\;B(y,r)\subset U\}$ and proofs it is continuous in this way: let $\epsilon>0$, there is some $x\in U$ such that $\epsilon(y)\leq|x-y|+\epsilon$. For any other $y'\in Y$, this is $\epsilon(y)\leq|x-y'|+|y'-y|+\epsilon$. Since $|x-y'|\leq\epsilon(y')$, we have $|\epsilon(y)-\epsilon(y')|\leq|y-y'|+\epsilon$.
I can't understand why $|x-y'|\leq\epsilon(y')$.
Remember that $U$ is a tubular neighborhood of $Y$ (not just an arbitrary open set containing $Y$). So if $|x-y'|=r$ and $x\in U$, then $B(y',r)\subset U$. It follows that if $\epsilon(y')<|x-y'|=r$, then $x\notin U$.