Summary: Math 311-001 201110
Assignment # 6 (due: April 13th)
1. Let
f(x, y) =



0 (x, y) = (0, 0)
x3
x2 + y2
(x, y) = (0, 0)
(a) Prove that D1f and D2f are bounded (hence f is continuous).
(b) Let : R R2
be continuously differentiable, with (0) = (0, 0)
and (0) = 0. Let g(t) = f((t)). Show that g is continuously
differentiable.
(c) In spite of the above, show that f is not differentiable at (0, 0).
2. Show that the continuity of f at a is necessary in the Inverse Function
Theorem, even when n = 1: let
f(t) =

Source: Argerami, Martin - Department of Mathematics and Statistics, University of Regina

Collections: Mathematics