Nonlinear Control Sy.. - Free

2.12. EXERCISES 61
(2.10) Show that a set A in a metric space (X, d) is closed if and only if its complement is
open.
(2.11) Let (X, d) be a metric space. Show that
(i) X and the empty set 0 are closed.
(ii) The intersection of any number of closed sets is closed.
(iii) The union of a finite collection of closed sets is closed.
(2.12) Let (X, dl) and (Y, d2) be metric spaces and consider a function f : X -* Y. Show
that f is continuous if and only if the inverse image of every open set in Y is open in
X.
(2.13) Determine the values of the following limits, whenever they exist, and determine
whether each function is continuous at (0, 0):
(1)
x2 _y2
x lOm0 1 + x2 + y2
(2.14) Consider the function
x
(ii) lim
x-*0,y-i0 x+ 2 Y22
x°+y`
for (x, y) # (0, 0)
0 for x = y = 0
(iii) X lim
(1 + y2) sin x
O X
Show that f (x, y) is not continuous (and thus not differentiable) at the origin. Proceed
as follows:
(i) Show that
lim f (x, y) = 0
x-40,y=x 0
that is, if x - 0 and y --> 0 along the parabola y = x2, then limx 0.
(ii) Show that
thus the result.
1
lira f (x, y) = -
x-+O,y=x-+0 2
(2.15) Given the function f (x, y) of exercise (2.13), show that the partial derivatives and
both exist at (0, 0). This shows that existence of the partial derivatives does not
imply continuity of the function.