Notes on metric spaces these notes introduce the concept of a metric space, which will be an essential notion throughout this course and in others that follow. Metric spaces, open balls, and limit points definition. Thishandoutshouldbereadbetweenchapters1and2ofthetext. Discrete data contains distinct or separate values. Open and closed sets in the discrete metric space mathonline. Many of the fundamental types of spaces used in analysis are metric spaces e. We warn the reader that we will later study a property of metric spaces called discreteness. Continuous functions in metric spaces throughout this section let x. Choose discretization and interpolation methods for converting between continuous and discrete models. Pdf fragmentability by the discrete metric researchgate. Introduction let x be an arbitrary set, which could consist of vectors in rn, functions, sequences, matrices, etc.
Continuous data is data that falls in a continuous sequence. In many cases a problem in a continuous framework can be handled by using a suitable method from discrete mathematics and conversely a beautiful description of such phenomena can be found in. Continuous functions between metric spaces the concept of continuous realvalued functions can be generalized to functions between metric spaces. Then this is a metric on xcalled the discrete metric and we call x. Function terminology examples i what is the range of this function. On the mathematical structure for discrete and continuous. Every function from discrete metric space to another metric space is uniformly continuous. It remains to establish the triangle inequality dx,y. Ninth grade lesson discrete and continuous functions. So, any function from a discrete metric space to any other metric space is uniformly continuous. As a formal logical statements, this theorem can be written in the following form.
It is clearly symmetric and nonnegative with dx,y0if and only if xy. Recall that any set m can be given the discrete metric d d given by d dx. If xy, then the left hand side is zero and the inequality certainly holds. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points.
There are many ways to make new metric spaces from old. For example, the step function 0, open balls, and limit points definition. Let c0, 1 be the set of all continuous rvalued functions on the interval 0, 1. Discrete data is countable while continuous data is measurable. Hamming distance let x be the set of all ordered triples of zeros and ones.
From continuous to discrete from discrete to continuous. Show that the discrete metric satisfies the properties of a metric. It is relatively easy to show that any interval in r is connected, and by the same argument that r itself is connected. Some methods tend to provide a better frequencydomain match between the original and converted systems, while others provide a better match in the time. Regrettably mathematical and statistical content in pdf files is unlikely to be accessible. Suppose that xis the continuous image of a separable metric space. Let x, ddisc be a metric space with the discrete metric. For homework you will show some a nice property of continuous functions on connected metric. Metrics on spaces of functions these metrics are important for many of the applications in analysis. Y is continuous if it is continuous at every point in x. Math 431 real analysis i homework due october 8 question 1. The natural metric on r is invariant with respect to translations, i. This is a metric on x which we call the discrete metric.
In particular, each singleton is an open set in the. Introduction to real analysis fall 2014 lecture notes vern i. Metric spaces are generalizations of the real line, in which some of the theorems that hold for r remain valid. M of all continuous functions from x into a metric space m, endowed with the topology of. Discrete data is the type of data that has clear spaces between values. Difference between discrete and continuous data with. The discrete topology is the finest topology that can be given on a set, i. In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. Introduction to real analysis fall 2014 lecture notes.
Sis called a neighbourhood of aif qcontains an epsilon. Co nite topology we declare that a subset u of r is open i either u. Continuous functions in a metric space using the discrete metric. Limits of functions in metric spaces yesterday we dened the limit of a sequence, and now we extend those ideas to functions from one metric space to another. A set endowed with the discrete metric is a discrete space, but there are discrete metric. Please suggest if it is correct and rigorous enough. Introduction when we consider properties of a reasonable function, probably the. Continuous functions on metric spaces uc davis mathematics. Any subset of r that is not an interval is not connected.
Further, the sequence a n 1 n is cauchy in this metric space but is not convergent. Continuous functions in a metric space using the discrete. The difference between discrete and continuous data can be drawn clearly on the following grounds. We will now look at the open and closed sets of a particular interesting example of a metric space the discrete metric space. Identity function continuous function between usual and discrete metric space. Two functions are close with respect to this metric if their values are close at every point of k. A set, whose elements we shall call points, is said to be a metric space if with any two points and of there is associated a real number, called the distance from to. A of open sets is called an open cover of x if every x. Some of the main results in real analysis are i cauchy sequences converge, ii for continuous functions flim n. The discrete metric is about the simplest possible notion of distance. This example shows how to convert a discretetime system to continuous time using d2c, and compares the results using two different interpolation methods. Quite often we need to know function values at any arbitrary point x.
It is not hard to check that d is a metric on x, usually referred to as the discrete metric. An epsilon neighbourhood of a2s is the set of all x2s such that dx. The following function on is continuous at every irrational point, and discontinuous at every rational point. Swbat determine if a function is discrete or continuous based on the situation that it models. If x,d is a metric space and a is a nonempty subset of x, we can make a metric d a on a by putting. The fact that every pair is spread out is why this metric is called discrete. Xthe number dx,y gives us the distance between them. By a neighbourhood of a point, we mean an open set containing that point.
Noncommutative geometry is the geometry of quantum space, which are generalized discrete spaces replacing classical quantum spaces woronowicz 1987, connes 1994. Suppose that ffngis a sequence of continuous functions converging to a function f. Some of this material is contained in optional sections of the book, but i will assume none of that and start from scratch. Y x is continuous if y is endowed with the discrete metric. Discrete data data tables experiment observations calculations continuous data analytics functions analytic solutions 3 from continuous to discrete soon after hurricane isabel september 2003 4 from discrete to continuous 5 what do we want from discrete sets of data. Is it clear to you that from this it follows that if x is discrete metric space, then every function on it is. R open and closed intervals, convergent sequences, continuous functions can be studied. In mathematics, a metric space is a set together with a metric on the set. Discrete and continuous random variables summer 2003.
Every function from discrete metric space to another metric space is uniformly continuous 4 is whether every topological space the continuous image of a metric space. A metric space is a set x equipped with a function called metric d x, that can be thought of as a measurement of the distance of any two elements in x. Y is uniformly continuous if for every 0 there exists 0 such that if x. Then the map is continuous as a function and check it. Every function from a discrete metric space is continuous at every point.
A subset uof a metric space xis closed if the complement xnuis open. Then this does define a metric, in which no distinct pair of points are close. Math 441 topology fall 2012 metric spaces by john m. Y is continuous at xif for every sequence fx ng that converges to x, the sequence ffx ngconverges to fx. A metric space is called complete if every cauchy sequence converges to a limit.