examples of continuous, bounded function

Question

What are the examples for a continuous vector valued function $f:\mathbb{R}^2\times \mathbb{R}\rightarrow \mathbb{R}^2$ which is also bounded on it's entire domain?

Answer 1

Pick any continuous from $\mathbb{R}^3$ to $\mathbb{R}^2$ whose image is fully contained inside some disk and it will do. There are infinitely many such functions.

As mentioned by David in the comments any constant function will do. For a slightly more complicated example consider the the following continuous function $f:\mathbb{R}^2 \to \mathbb{R}^2$

$$f(x,y) = \begin{cases} \langle x,y \rangle & x^2+y^2\leq 1 \\ \frac{\langle x,y \rangle}{\sqrt{x^2+y^2}} & x^2+y^2>1 \end{cases} $$

Since $f$ maps the whole $\mathbb{R}^2$ into the unit disk, composing it with any continuous function from $\mathbb{R}^2\times\mathbb{R}$ to $\mathbb{R}^2$ will yield a continuous bounded function.

For example if $g_1(\langle x,y \rangle,z)= \langle x,y \rangle$ then $f\circ g_1$ will be continuous and bounded. For other functions to compose you can try:

\begin{align} g_2(\langle x,y \rangle,z) &= \langle x+z,y-z \rangle\\ g_3(\langle x,y \rangle,z) &= \langle xz,yz \rangle\\ g_4(\langle x,y \rangle,z) &= \langle y,-x \rangle\\ g_5(\langle x,y \rangle,z) &= \langle e^{x+y}\cos{z}, e^{x-y}\sin{z} \rangle\\\end{align}