I want to check continuity of map $f:\mathbb{R^2} \to D^2$ (where $D^2$ is closed unit ball) given by:
$$x \to\begin{cases} x & \|x\|\leq 1 \\ \frac{x}{\|x\|} & \|x\|>1 \end{cases}$$
My attempt:
I am wondering if it is enough to show that, $ \lim_{||x||\to 1} f(x)=x $. That seems to be true intuitively. I need a rigorous proof.
Any help will be greatly appreciated!!
The pasting lemma is what you are looking for here. https://en.wikipedia.org/wiki/Pasting_lemma
Let $X$ be the closed unit ball and let $Y$ be $\mathbb{R}^2$ minus the open unit ball, these are both closed spaces and their union is all of $\mathbb{R}^2$. So to use the pasting lemma, we just need $f$ to have unambiguous definitions on their boundary, i.e. it wouldn't matter which formula you used. Fortunately you have that here. So we can say that $f$ restricted to $X$ is $f(x)=x$, continuous, and $f$ restricted to $Y$ is $f(x)=\frac x {\|x\|}$, continuous. Note this only works because on the overlap the two definitions agree. Then the pasting lemma gives us the function itself is continuous