Let $f:A\mapsto\mathbb{R}$ be a function. Let $A = A_1\cup A_2 \cup\ldots$ is an infinite decomposition of the domain $A$ of $f$ such that for some $\delta_0>0$ and $i\ne j\in\mathbb{N}$, we have $|x-y| \ge \delta_0$ for all $x\in A_i$ and $y\in A_j$.
Show that if $f$ is uniformly continuous on each of the sets $A_i$ independently, Is it true that f is uniformly continuous on $A$ as well? why?
To begin with, we need to use the uniform continuity definition, but what is the next step, did someone give me some clue on it?
This is false. For example let $A_n=[2n,2n+1]$ and $f(x)=x^{2}$. Then the hypothesis holds with $\delta_0=1$ but $f$ is not uniformly continuous on $A$.