Convergence & Continuity

Transcription

1 September 5 8, 217

2 Given a sequence {x n } n Ξ of elements x n X, where X is a metric space, Ξ is an indexing set. Let Ξ = N Definition (convergent sequence) A sequence {x n } is called convergent sequence iff there is a point x s.t. ρ(x n,x) as n, or equivalently, ε > N N s.t. n N : ρ(x n,x) < ε Then, x is called a limit of {x n }, and we say that {x n } X converges to x X and write x n x or lim n x n = x

3 (cont.) Example: X = C[,1] with ρ 2 (, ). Consider the following {x n } in (C[,1],ρ 2 ): { 1 nt, t 1/n x n (t) =, 1/n < t 1 Claim: x n x. Indeed, [ [ 1 ]1 2 ]1 1/n 2 ρ 2(x,y) = x n(t) x (t) 2 dt = (1 nt) 2 dt = (3n) 1 2. n Thus, given ε > if n N where N N > 1 ε 2 then ρ 2(x n,) < ε Theorem A sequence {x n } of a metric space may converge to only one limit It is clear that a sequence {x n } is convergent to x iff ε > B(x,ε) contains all but a finite number of the x n

4 (cont.) Let {x n } X be a sequence in X. Then lim supx n := lim (sup{x k : k n}) =: lim x n n n lim inf n x n := lim n (inf{x k : k n}) =: lim x n Note: x n converges iff liminf x n = limsup x n Is is also clear that x X is a point of accumulation of M iff {x n } M \{x} s.t. x n x as n Theorem (Closed Set Theorem) A set M in a metric space (X,ρ) is a closed set iff every convergent sequence {x n } M has its limit in M

5 (cont.) Let (X,ρ X ), (Y,ρ Y ) are metric spaces. Consider a function (map, operator) f mapping the metric space (X,ρ X ) to a metric space (Y,ρ Y ). Function f is defined for x D X, then D is called a domain of f and f(d) is its range Definition (mapping continuous at a point) A function f(x) : D X Y defined on some set D X with range in Y is continuous at x D iff ε > δ > s.t. ρ X (x,x ) < δ ρ Y (f(x),f(x )) < ε. Hence, x n x (x n,x X) f(x n) f(x ) Definition (continuous and discontinuous mapping) A function f(x) is continuous on M X iff it is continuous at every point in M. If f is not continuous it is discontinuous Theorem Given f : (X,ρ X ) (Y,ρ Y ). Let x X. Then f is continuous at x iff for every sequence {x n} X s.t. lim xn = x one has lim f(xn) = f( lim xn) n n n

6 (cont.) Examples: 1 X = Y = L 2[,T], < T <, with usual metric. Then y(t) = t x(s) ds, represents a mapping f of L 2[,T] into itself. This f is continuous. Moreover, uniformly continuous (see def n below) 2 Let Y = L 2(, ) with usual metric ( ) 1/2 ρ(x,y) = x(t) y(t) 2 dt and let X be the subspace of Y made up of all points x Y s.t. t 2 x(s)ds dt < Define a mapping f of X into Y by It is not continuous. y(t) = t x(s) ds

7 Examples (cont.): 3 Consider F : X = C[,T] R defined by x(t)dt, x X. Then F is continuous mapping in (C[,T],ρ ) since F(x) F(y) = x(t)dt y(t)dt x(t) y(t) dt Cρ (x,y) with C = T, hence, continuity holds. We also know that if {x n } X and if x n x in (X,ρ ) x n (t) converges uniformly to x (t), thus lim n x n (t)dt = provided x n converges uniformly. lim x n(t)dt n

8 Uniform Continuity Yuliya Gorb Definition (uniformly continuous mapping) A mapping f of a metric space (X,ρ X ) into a metric space (Y,ρ Y ) is uniformly continuous iff ε > δ > s.t. x,y X : ρ X (x,y) < δ ρ Y (f(x),f(y)) < ε Here δ depends on ε only, but not on x! Example: f(x) = logx is continuous on (,1) but not uniformly continuous there, i.e. we have to show that ε > δ > x,y X : ρ X (x,y) < δ ρ Y (f(x),f(y)) ε Choose ε = 1 and any < δ < 2(e 1) s.t. ρ X (x,y) < δ x > and y = x +δ/2, thus, logx logy = logx log(x +δ/2) = log(1+ δ 2x ) > 1 =: ε, δ hence, < x < 2(e 1)

9 Upper and Lower Semicontinuity Definition (upper semicontinuous mapping) A mapping f : X R is upper semicontinuous on X iff x X, x n X : x n x limsupf(x n ) f(x) n Definition (lower semicontinuous mapping) A mapping f : X R is lower semicontinuous on X iff x X, x n X : x n x liminf n f(x n) f(x) Hence, f is continuous iff f is both upper and lower semicontinuous

10 Lipschitz Continuity Yuliya Gorb Definition (Lipschitz continuous mapping) A mapping f of a metric space (X,ρ X ) into a metric space (Y,ρ Y ) is Lipschitz continuous on X iff L > s.t. ρ Y (f(x),f(y)) Lρ X (x,y), x,y X Definition (Lipschitz constant) If f : X Y is Lipschitz continuous on X then define the Lipschitz constant of f by ρ Y (f(x),f(y)) Lip f := sup x y ρ X (x,y) Equivalently, Lip f is a smallest constant L in definition: Lip f := inf {L : ρ Y (f(x),f(y)) Lρ X (x,y), x,y X} Note: every Lipschitz continuous function is uniformly continuous but there are uniformly continuous functions that are not Lipschitz continuous. Example: f(x) = x is uniformly continuous but not Lipschitz continuous in [,1] since f(x) f() x lim = lim x + x x + x =

11 References Yuliya Gorb Hunter/Nachtergaele Applied Analysis pp. 6 9, 11 14, Naylor/Sell Lineat Operator Theory... pp