Afunction f: r^{^{n}}\rightarrow r^{m} ? is said to be proper if it is continuous and the following implication holds:
for any sequence x? k , \left \| x_{k} \right \|\rightarrow \infty \rightarrow \left \| f(x_{k}) \right \|\rightarrow \infty .
a) give an example of a proper function from r to r.
b) show that a continuous function from r? n to rm ? is proper if and only if inverse image of any compact set is compact.
c) if f: rn \rightarrow ? r is a proper function, show that |f| attains its global minimum on rn.