Is this piecewise function continuous? Is the above function continuous? Intuitively, I think the answer is no because at points where the norm is equal to 1 and points where the norm is slightly larger than 1, it feels like there is a &quot;jump discontinuity&quot; but I'm not sure how to show that.
Yes, it is continuous. It is the product of two continuous functions: the identity function and the function$$\begin{array}{rccc}a\colon&amp;\Bbb R^n&amp;\longrightarrow&amp;\Bbb R\\&amp;x&amp;\mapsto&amp;\begin{cases}1&amp;\text{ if }\|x\|&lt;1\\\frac1{\|x\|}&amp;\text{ otherwise.}\end{cases}&amp;\end{array}$$The function $a$ is continuous because, for each $x\in\Bbb R^n\setminus\{0\}$,$$a(x)=\min\left\{1,\frac1{\|x\|}\right\}=\frac{1+\frac1{\|x\|}-\left|1-\frac1{\|x\|}\right|}2.$$
For all $y$ such that $\|y\|=1$,

for all $x:\|x\|&gt;1$, $$\lim_{x\to y}f(x)=\lim_{x\to y}\frac{x}{\|x\|}=\frac{\lim_{x\to y}x}{\lim_{x\to y}\|x\|}=\frac y1=f(y).$$

for all $x:\|x\|\le1$, $$\lim_{x\to y}x=y=f(y).$$