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Continuous Functions

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Metric Space

Let V be a vector space, a #~{metric} on V is a function _ d#: V # V --> &reals.^+ _ such that:

  1. d( ~x, ~y ) _ >= _ 0 _ _ &forall. ~x, ~y
  2. d( ~x, ~y ) _ = _ 0 _ _ <=> _ _ ~x = ~y
  3. d( ~x, ~y ) _ = _ d( ~y, ~x ) _ _ &forall. ~x, ~y
  4. d( ~x, ~z ) _ =< _ d( ~x, ~y ) + d( ~y, ~z ) _ _ (triangle inequality)

V is then said to be a _ #~{metric space} _ with metric d, and is represented by the pair (V, d)

Closed Ball

(V, d) is a metric space. The _ #~{closed ball} _ in V with centre _ ~a _ radius _ ~r _ _ (~r &in. &reals.) _ is _ &set. ~x &in. V | d( ~a , ~x ) =< ~r &xset. _ denoted _ B[ ~a , ~r ]

Open Ball

The _ #~{open ball} _ in V with centre _ ~a _ radius _ ~r _ _ (~r &in. &reals.), _ is _ &set. ~x &in. V | d( ~a , ~x ) < ~r &xset. _ denoted _ B( ~a , ~r )

Continuous Function

Let _ (A, d_A) _ and _ (B, d_B) be metric spaces. (We will write 'd' for both 'd_A' and 'd_B', it should be clear from context which is which).
f#: A --> B _ is a #~{continuous function} at a point ~a &in. A _ if given &epsilon. > 0 _ &exist. &delta. > 0 such that _ d( ~a, ~x ) < &delta. _ => _ d( f ( ~a ), f ( ~x ) ) < &epsilon. .

f is said to be #~{continuous (on A)} if it is continuous at every point of A.

Composite of Continuous Functions

If _ f#: A --> B _ and _ g#: B --> C _ are continuous, then _ g &comp. f #: A --> C is also continuous.

Proof:
Let _ f ( ~a ) = ~b _ for some ~a &in. A. Then given &epsilon. > 0 , _ &exist. &eta. > 0 _ such that _ d( g(~b), g ( ~x ) ) < &epsilon. _ if _ d( ~b, ~x ) < &eta. .
In particular if ~x is such that _ ~x = f ( ~y) _ some ~y &in. A , _ then &exist. &delta. > 0 _ such that _ d( f ( ~a ), f ( ~y) ) < &eta. _ if _ d( ~a, ~y ) < &delta. .
So given &epsilon. > 0 _ we have found &delta. > 0 _ such that _ d( (g &comp. f)( ~a ), (g &comp. f)( ~y ) ) < &epsilon. _ if _ d( ~a, ~y ) < &delta..

Real and Complex Number Space

In the sequel we will consider functions whose co-domain is either &reals.^~n or &complex.^~n with the usual metric,
_ _ _ _ _ i.e. _ d( ~x , ~y ) _ = _ | ~x - ~y | _ = _ &sqrt.( &sum._~i (~x_~i - ~y_~i)^~n ) _ _ ~x , ~y &in. &reals.^~n _ ( similarly &complex.^~n ).

In particular we have _ | ~x - ~y | _ = _ | (~x + ~a) - ( ~y + ~a) | _ _ ("parallel shift").

Continuity of Sum and Product

If _ f#: A --> &reals. _ and _ g#: A --> &reals. _ are continuous then so are

  1. f + g #: A --> &reals.
  2. fg #: A --> &reals.

Proof:

  1. | (f ( ~x ) + g ( ~x )) - (f ( ~a ) + g ( ~a )) | _ = _ | (f ( ~x ) - f ( ~a )) + (g ( ~x ) - g ( ~a )) |
    _ _ _ =< _ | f ( ~x ) - f ( ~a ) | + | g ( ~x ) - g ( ~a ) | _ _ (triangle inequality)
    f and g are continuous, _ &therefore. given &epsilon. > 0 _ &exist. &delta._1 , &delta._2 _ such that
    | f ( ~x ) - f ( ~a ) | < &epsilon./2 _ if _ d(~x,~a) < &delta._1 _ and _ | g ( ~x ) - g ( ~a ) | < &epsilon./2 _ if _ d(~x,~a) < &delta._2
    put _ &delta. = min&set. &delta._1 , &delta._2 &xset. , _ so | (f ( ~x ) + g ( ~x )) - (f ( ~a ) + g ( ~a )) | _ < _ &epsilon. _ if _ d(~x,~a) < &delta. .
    _
  2. given &epsilon._0 = 1

 

 

 

Continuity of Linear, Constant and Reciprocal Functions

  1. g#: &reals. --> &reals. _ _ g ( ~x ) = ~x _ is continuous on &reals.
  2. g#: &reals. --> &reals. _ _ g ( ~x ) = ~c _ (constant) _ is continuous on &reals.
  3. g#: &reals. --> &reals. _ _ g ( ~x ) = 1/~x _ is continuous on &reals. 0

Proof:

  1. Trivial.
    _
  2. Trivial.
    _
  3. To be provided
    _