234 CHAPTER 8. IMPLICIT FUNCTION THEOREM

25. ↑ Show that if f :Rn→Rn is C1 and if Df (x) exists and is invertible for all x∈Rn,then f is locally one to one. Thus, from the above problem, if lim|x|→∞ |f (x)|= ∞,

then f is also onto. Now consider f : R2→ R2 given by

f (x,y) =(

ex cosyex siny

)Show that this does not map onto R2. In fact, it fails to hit (0,0), but Df (x,y) isinvertible for all (x,y). Show why it fails to satisfy the limit condition.

26. You know from linear algebra that there is no onto linear mapping A : Rm→ Rp forp > m. Show that there is no locally one to one continuous mapping which will mapRm onto Rp.

27. In Example 8.1.9 on Page 210, could you replace y with y ∈ Rm and obtain a modi-fied version of this example?