Principles of Mathematics in Operations Research

158 11 Continuity

Ve > 0,36 > 0 9 Vx 6 E with dx(x,p) < S we have du(f(x), f(p)) < s.

Remark 11.1.3 The following characteristics are noted:

• / has to be defined at p, but p does not need to be a limit point of E.


• Ifp is an isolated point ofE, then f is continuous at p. That is, given e > 0
  (no matter what £ is), find S 3 E fl Bf{p) = {p}. Then, x & E, d{p, x) <
  6 =>• x = p. Hence, dy(f(x), f(p)) = 0 < e.


• Ifp is a limit point of E, then f is continuous atp-^f- limx_>p /(x) = /(p).

Definition 11.1.4 If f is continuous at every point of E, we say f is con-

tinuous on E.

Proposition 11.1.5 Let (X,dx),(Y,dY),(Z,dz) be metric spaces and 0 ^

E C X, f : E i-» Y, g : f(E) i-» Z. If f is continuous at p G E and g is

continuous at f(p), then g o f is continuous at p.

Proof. Let q = /(p). Let e > 0 be given. Since g is continuous at q, 3n > 0 9

Vy G f(E) with dy(y,q) < n we have d-2(g{y),g(q)) < e. Since / is continuous

at p, 35 > 0 9 Vx e E with dx{x,p) < 6 =$> we have dy(/(x),/(p)) < n.

Let x G E be 9 djr(:r,p) < <5. Then, y = f(x) G /(E) and dY(y,q) =

dy(f(x),f(p)) < n. Hence, dz{g{f{x)),g{f{p))) = dz(g(y),g(q)) < s. •

Theorem 11.1.6 Lei (X, dx-),(Y, dy) 6e metric spaces, and let f : X t-> Y.

T/ien, / is continuous on X if and only i/V open se£ V in Y, /-1(V) ={p€

•^ : /(p) S y} is open in X.

Proof. (=>•):

Let V be open in Y. If f~l(V) ^ 0, let p G /_1(^) be arbitrary. Show

3r > 0 9 B*(p) C /-1(^): p G /-1(V) implies /(p) G V. Since V is open,

3s > 0 9 BY(f(p)) C V. Since / is continuous at p, for e = s, 3r > 0 9 Vx G

X with 4(x,p) < r =• <W(x), /(p)) < s =* /(x) G Bsy(/(p)) =» x G /"^(V).

Let p G X be arbitrary. Given e > 0, let V = J3^(/(p)) be open. Then,
/-1(V) is open and p G f~l{V). Hence, 3<J 9 B«(p) C /_1(V). If <4(x,p) <
5 => x G Bf (p) C /"HV), then /(x) G V => rf,(/(x),/(p)) < e. D

Corollary 11.1.7 / : X —> Y is continuous on X if and only i/V closed set
C in Y, f~l{C) is closed in X.

Proof. f-l{Ec) = {f-\E))c. U

Definition 11.1.8 Let (X,d) be a metric space and /:,...,/* : X i-» R.
De/me / : X M- Rfc 6j/ /(x) = (/i(x),..., fk{x))T, then /i,..., /fc are ca/ted
components of f.