Originally Posted by

**zebra2147** Attempt #1 at proof:

Suppose $\displaystyle D=D_{1}\bigcup D_{2}$ where $\displaystyle D_{1}$ contains only negative real numbers and $\displaystyle D_{2}$ contains only positive real values. Let $\displaystyle a$ be a limit point of $\displaystyle D_{1}$ and $\displaystyle D_{2}$. Then let $\displaystyle f: D\rightarrow R$ be the function $\displaystyle f(B'_{r}(a))$. Thus, the $\displaystyle lim_{x\longrightarrow a}$ exists only if $\displaystyle lim_{x\longrightarrow a}f|_D_{j}$ both exist and are equal.

Then, if both limits exist we have that, for some $\displaystyle \epsilon>0$,when $\displaystyle x_{1}<0$, $\displaystyle |x_{1}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$. Also, when $\displaystyle x_{2}>0$, $\displaystyle |x_{2}-a|<\delta\Rightarrow |f(B'_{r}(a))-b|<\epsilon$.

The only way that this can happen is when $\displaystyle b=0$?????