ANALYSIS II
Metric Spaces: Limits and Continuity

Defn   Suppose (X,d) is a metric space and A is a subset of X.

  1. A point x is called an interior point of A if there is a neighborhood of x contained in A.
  2. A set N is called a neighborhood (nbhd) of x if x is an interior point of N.
  3. A point x is called a boundary point of A if it is a limit point of both A and its complement.
  4. A point x is called a limit point of the set A if each neighborhood of x contains points of A distinct from x.
    (This is equivalent to saying that each neighborhood of x has an infinite number of members of A.  Recall that a neighborhood for a point x, is a set containing an open -nbhd of x.)
  5. A point x is called an isolated point of A if x belongs to A but is not a limit point of A.

Proposition  A set  O  in a metric space is open if and only if each of its points are interior points.

Proposition  A set  C  in a metric space is closed if and only if it contains all its limit points.

Defn   Suppose (X,d) is a metric space and A is a subset of X. The closure of A is the smallest closed subset of X which contains A. The derived set A' of A is the set of all limit points of A.

Proposition  The closure of A may be determined by either

or

Sequential Convergence

Defn   A sequence {xn} in a metric space (X,d) is said to converge, to a point x0 say, if for each neighborhood of x0 there exists a natural number N so that xn belongs to the neighborhood if n is greater or equal to N; that is, eventually the sequence is contained in the neighborhood. In this case, we say that x0 is the limit of the sequence and write
                         xn := x0 .

Proposition  In a metric space, sequential limits are unique.

Proposition  That a sequence {xn} converges in a metric space (X,d) to a point x0 is equivalent to the condition that for each  > 0 there is a natural number N such that N  n implies d(xn , x0) <  .

Examples 

  1. In either the reals or complexes if |r| < 1, then rn  0.
  2. Consider the space of continuous functions on [0,1/2], C[0,1/2]. Let fn(x) = xn, then fn  0.
  3. The sequence fn(x) = xn  belongs to C[0,1] but does not converge.

Defn   A  function f  defined on X\{x0}, with values in a metric space {Y,d2} is said to have a limit L at x0 if x0 is a limit point of X and for each neighborhood O2 of L, there is a neighborhood O1 of x0 such that f maps each element of the deleted neighborhood O1\{x0} into O2 . This is denoted

                           f(x) := L.

Homework This is equivalent to the condition: for each   > 0  there is a   > 0  such that if  0 < d1(x,x0) < , then d2(f(x),L) < .

Proposition  A necessary and sufficient condition for a function f to have a limit L at x0 is that for each sequence {xn} which converges to x0 (no point of which is equal to x0), then {f(xn)} converges to L. Consequently, if a function has a limit at a point x0, then it is unique.

Defn   A function f  is called continuous at a point x0 if either

  1. x0 is an isolated point of X or
  2. x0 is a limit point of X and the limit of f as x approaches x0 is f(x0).

Homework  A necessary and sufficient condition for a function f to be continuous at x0 is that for each   > 0  there is a   > 0  such that if   d1(x,x0) < , then d2(f(x),f(x0)) < .

Continuity

Defn  Suppose f : X  Y where (X,d1) and (Y,d2) are metric spaces.  f  is called continuous if  the inverse image of each open set in Y is open in X.

Proposition  A function f : X  Y is continuous if and only if the inverse image of each closed set in Y is closed in X.

Theorem  A function f : X  Y is continuous if and only if f is continuous at each point of X.

Theorem  Suppose that f: X  Y and g: Y  Z are continuous functions, then gof is a continuous function from X to Z.

Theorem  Suppose that (X,dX) and  (Y,dY) are both metric spaces, then X x Y is a metric space if the metric d is defined for zi = (xi,yi), i=1,2, by

d(z1,z2) := dX(x1,x2) + dY(y1,y2).

Examples:  

  1. For a metric space (X,d), the metric d is a continuous function from X2 to R.
  2. Suppose that (X,||.||) is a normed linear space, then both the vector space operations are jointly continuous:
    1. if an  a in R and xn x in X, then ||an xn||X  ||a x||X in R.
    2. if xn x  and  yn y  in X, then   xn + yn    x +y   in X.

Robert Sharpley Jan 25 1998