Basic Continuity

Continuity is perhaps one of the most fundamental topics in all of analysis, as it provides a way of specifying mappings that are nicely behaved and suitable for studying. Many topics in mathematics (i.e. a homeomorphism) rely on the underlying notion of continuity. Continuous functions have rather nice properties, such as being Riemann and Lebesgue Integrable, mapping compact sets into compact sets, and being uniquely determined by their values on a dense subset.

I begin with the introduction of a metric space, discuss continuity in these spaces, and thus provide motivation for the development of topological spaces.

Metric Spaces

Definition 1:  Let X be a set. A Metric Space is simply a pair (X,d) where d is known as the metric function so that d: X \times X \to \mathbb{R}^+_0 and satisfies the following conditions for x,y \in X:

  • d is non-negative, with d(x,y) = 0 if and only if x=y.
  • d is symmetric: d(x,y) = d(y,x)
  • d(x,z) \leq d(x,y) + d(y,z) (Triangle Inequality)

Thus, a metric space is simply a space where there is a well-defined notion of “how close” points are, so that the points are topologically distinguishable (in fact, a metric space is Hausdorff).

For example, we may equip \mathbb{R}^n with a metric d so that (\mathbb{R}^n,d) is a metric space. Often, we will use one of the following two metrics for x,y \in \mathbb{R}^n:

d_1(x,y) = ||x-y|| = \Big( \sum \limits_{i=1}^{n} (x_i - y_i)^2 \Big)^{\frac{1}{2}} called the Euclidean Metric (as it is derived from inductively applying the Pythagorean Theorem). As well as the supremum metric:

d_2(x,y) = ||x-y||_\infty = \sup_{i} |x_i - y_i| = \max_{i} |x_i -y_i| since \mathbb{R}^n is finite-dimensional.

The ambiguity as to which metric we should assign to a space therefore arises, as these seem to be entirely different ways of equipping a space with a metric. Consequently, we have the following definition:

Definition 2: Let (X,d_1) and (X,d_2) be metric spaces. We define an equivalence relation d_1 \sim d_2 if d_1(x,y) \leq Md_2(x,y) for all x,y \in X and for some M \in \mathbb{R}.

Now, it may be (simple exercise) shown that ||x-y||_\infty \leq ||x-y|| \leq n^{\frac{1}{2}} ||x-y||_\infty. Thus, there is a notion of equivalence amongst metrics, and so we may interchange them without losing essential information about these sets.

Another basic, though less obvious, example of a metric is the discrete metric. This is the metric space (X,d) where d is defined to be d=\left\{  \begin{array}{lr}  0 & : x =y \\  1 & : x \neq y  \end{array}  \right.. As we’ll see in a moment, this is an interesting metric space, as all functions with this function as domain will be continuous.

Metric spaces also have notions of open and closed sets. Define an open ball (or neighborhood) of radius \epsilon centered at x \in X to be the set B_\epsilon(x) := \{ y \in X : d(x,y) < r\}. (Equivalently, define a closed ball to be the set B_\epsilon(x) := \{ y \in X : d(x,y) \leq r \}). Using these, we assume the reader has knowledge of open and closed sets in a metric space. Namely, A \subset X is open in X if for each a \in A there exists \epsilon >0 so that B_\epsilon(a) \subset A. Equivalently, A \subset X is closed in X if A contains its limit points. If the reader is not familiar with this notion, check the notion of convergence in a Metric Space.

Continuity in Metric Spaces

We now introduce the notion of continuity in metric spaces:

Definition 3: Let (X,d_X) and (Y,d_Y) be metric spaces. We say that f: X \to Y is continuous at x \in X if, for every \epsilon >0, there exists \delta > 0 so that if d_X(x,y) < \delta then d_Y(f(x),f(y)) < \epsilon. If f is continuous at each x \in X, we say that f is continuous on X.

Intuitively, this means that if two points in X are “close enough” we can ensure that two points in the co-domain of f are also “close”.

In fact, a metric is itself a continuous function. Fix a \in X. We shall show first d(a,x) is continuous as a function d_a: X \to \mathbb{R}^{+}_0. We have |d(a,x) - d(a,y)| \leq d(x,y) if we apply the triangle inequality. Thus, choose \delta = \epsilon and we see that if d(x,y) < \delta then |d(a,x) - d(a,y)| < \epsilon, so that d(a,x) is continuous. Now, reverse the roles of a and x. Combining these facts, we see that, for (x,y),(x_0,y_0) \in X \times X, where d(x,x_0) < \frac{\epsilon}{2} and d(y,y_0) < \frac{\epsilon}{2},

|d(x,y) - d(x_0,y_0)| \leq |d(x,y)-d(y,x_0)| + |d(y,x_0) - d(x_0,y_0)|

by the triangle in equality we therefore have

|d(x,y)-d(y,x_0)| + |d(y,x_0) - d(x_0,y_0)| \leq d(x,x_0) + d(y,y_0) < \epsilon

so that the metric function is indeed continuous.

Another important form of a continuous function (in fact, it is uniformly continuous) is that of a Lipschitz function. This is a function f: X \to Y such that d_Y(f(x),f(y)) \leq Ld_X(x,y). To see continuity, choose \delta = \frac{\epsilon}{L}. For those familiar with the notion, it should be noted that bijectivity and continuity do not ensure that a mapping is a homeomorphism (a mapping f so that f is continuous, bijective, and its inverse continuous). To see this, let (\mathbb{R}^n,d) and (\mathbb{R}^n,d_{*}) be the metric spaces equipped with the Euclidean and Discrete metics, respectively. Then, the inclusion map

\iota: (\mathbb{R}^n,d_{*}) \to(\mathbb{R}^n,d)

is continuous and bijective, yet

\iota^{-1} : (\mathbb{R}^n,d) \to (\mathbb{R}^n,d_{*})

fails to be continuous.

Characterizations of Continuity in Metric Spaces

A nice property of metric spaces is the fact that we may characterize the notion of continuity of a mapping f in four different, yet equivalent ways.

Theorem 1: Let (X,d_X) and (Y,d_Y) be metric spaces. Then, the following are equivalent statements:

  1. f is continuous at a \in X.
  2. \epsilon-\delta requirement of Definition 3.
  3. Suppose U \subset Y is open in Y. Then, f^{-1}(U) is open in X.
  4. Suppose E \subset Y is closed in Y. Then, f^{-1}(E) is closed in X.
  5. (requires notion of convergence) If \{x_n\}_{n \in \mathbb{N}} \to x, then \{f(x_n)\}_{n \in \mathbb{N}} \to f(x).

Proof: (1) \implies (2) by definition. To see (2) \implies (3), we note that d(f(a),f(y)) < \epsilon is precisely the definition of the open ball centered at f(a) with radius \epsilon. Thus, we have \delta so that f^{-1}(B_\epsilon (f(a))) \subset B_\delta (a) so that the inverse image is open. To see (3) \implies (4), we simply take complements. A set is closed if and only if its complement is open. Thus, Y \setminus E is open and so f^{-1}(Y \setminus E) = f^{-1}(Y) \setminus f^{-1}(E) is open. Thus, X \setminus (f^{-1}(Y) \setminus f^{-1}(E)) = f^{-1}(E) is closed. The other directions as well as (5) are left to the reader as simple exercises.

Continuity as a Motivation for Topological Spaces

Suppose we wish to move away from the notion of a metric space to a more general space, called a topological space. I want my new space to be one in which there is a well-defined notion of continuity. How then, should I define this new space? Well, why not make use of (3) in Theorem 1? This gives that the only notion required for continuity is that inverse images of open sets are open. Thus, it makes sense for our new space to have some sense of an open set. We then ask: How do open sets behave in \mathbb{R}? Let \mathscr{O} be the collection of all open subsets of \mathbb{R}. Then, the following can be easily checked:

  • \emptyset, \mathbb{R} \in \mathscr{O}.
  • For a finite index set I, and open sets \{V_i\}_{i \in I}, \bigcap \limits_{i \in I} V_i \in \mathscr{O}.
  • For an arbitrary index set J, and open sets \{U_j\}_{j \in J}, \bigcup \limits_{j \in J} V_j \in \mathscr{O}

Ahh, you say. Why not then define a space that behaves in this manner? A collection of sets that we will call open sets that behave in the same way \mathscr{O} does. Well, this is precisely the motivation behind the definition of a topology.

Let X be a set. We define a topology on X to be a collection of sets \mathscr{T} whose members we will call open sets, satisfying:

  • \emptyset, X \in \mathscr{T}.
  • For a finite index set I, and open sets \{V_i\}_{i \in I}, \bigcap \limits_{i \in I} V_i \in \mathscr{T}.
  • For an arbitrary index set J, and open sets \{U_j\}_{j \in J}, \bigcup \limits_{j \in J} U_j \in \mathscr{T}

We call (X,\mathscr{T}) a topological space.

This gives a natural definition of continuity in these spaces:

Definition 4: Let (X,\mathscr{T}) and (Y, \mathscr{V}) be topological spaces. Let f: (X, \mathscr{T}) \to (Y, \mathscr{V}). We say that f is continuous at a \in X if, for every open neighborhood V \in \mathscr{V} of f(a), there exists an open neighborhood U \in \mathscr{T} of a so that f(x) \in V whenever x \in U. If f is continuous at every a \in X, we say that f is continuous on X.

That’s all for now, hopefully this entry gives some insight as to the motivation behind topological spaces, as well as an introduction of the basic concept of continuity. As usual, comments are welcome and all inquiries should be directed to erdosninth@gmail.com

Cheers,

J.T. (S.D.)

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