If we have $\lim f(x) = b$ and $f:I\to\Bbb{R}$ , does $\lim f(x)=b$ for $f:\Bbb{R}\to\Bbb{R}$?

Question

If we have $\lim_{x\to a} f(x) = b$ and $f:I\to\Bbb{R}$ , does $\lim_{x\to a} f(x)=b$ for $f:\Bbb{R}\to\Bbb{R}$? and vice versa?

It seems obvious but i can't prove it using the epsilon - delta definition

Answer 1

Actually no, it is false. Let $I=(0,1)$, $f(x)=1$ in $I$, $f(x)=0$ outside $I$. Then $\lim_{x\to 0} f(x)=1$ if considered as $f:I\to\mathbb{R}$, limit does not exist if considered as $f:\mathbb{R}\to\mathbb{R}$.