Metric space property of balls

Is it true that for a complete metric space $X$, for every continuous function $\epsilon:X\rightarrow (0,\infty)$, there exists a continuous function $\delta:X\rightarrow (0,\infty)$ such that for any $x'\in X$ if $x\in B_{\epsilon(x')}(x')$ then $B_{\delta(x)}(x)\subseteq B_{\epsilon(x')}(x')$?

Ideas:As $B_{\epsilon(x')}(x')$ is open, for any $x\in X$, there exists $\delta_x>0$ such that $B_{\delta_x}(x)\subseteq B_{\epsilon(x')}(x')$. But $\delta:B_{\epsilon(x')}(x')\rightarrow (0,\infty)$ given by $\delta(x)=\delta_x$ is only defined locally and don't have to be continuous. If $\delta$ is continuous, then maybe partitions of unity will give what we want?