Continuity of F G in Metric Space

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ANALYSIS II
Metric Spaces: Limits and Continuity

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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

• the intersection of all closed sets which contain A,

or

• the union of A with its derived set.

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Sequential Convergence

Defn A sequence {x_n} 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 x_n 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
x_n := x_{0 .}

Proposition In a metric space, sequential limits are unique.

Proposition That a sequence {x_n} 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(x_n , 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 f_n(x) = xⁿ, then f_n  0.
3. The sequence f_n(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 {x_n} which converges to x₀ (no point of which is equal to x₀), then {f(x_n)} 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)) < .

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Continuity

Defn Suppose f : X  Y where (X,d ₁ ) and (Y,d ₂ ) 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 g o f is a continuous function from X to Z.

Theorem Suppose that (X,d _X ) and  (Y,d _Y ) are both metric spaces, then X x Y is a metric space if the metric d is defined for z_i = (x_i,y_i), i=1,2, by

d(z₁,z₂) := d _X (x₁,x₂) + d _Y (y₁,y₂).

Examples:

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

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Robert Sharpley Jan 25 1998

arnetterepasustem.blogspot.com

Source: https://people.math.sc.edu/sharpley/math555/Lectures/MetricSpaceLimits.html

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