Basic Analysis IPub Date : 2020-05-13DOI: 10.1201/9781315166254-15
J. Peterson
{"title":"Exponential and Logarithm Functions","authors":"J. Peterson","doi":"10.1201/9781315166254-15","DOIUrl":"https://doi.org/10.1201/9781315166254-15","url":null,"abstract":"Exponential functions and logarithm functions are important in both theory and practice. In this unit we look at the graphs of exponential and logarithm functions, and see how they are related. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: • specify for which values of a the exponential function f (x) = a x may be defined, • recognize the domain and range of an exponential function, • identify a particular point which is on the graph of every exponential function, • specify for which values of a the logarithm function f (x) = log a x may be defined, • recognize the domain and range of a logarithm function, • identify a particular point which is on the graph of every logarithm function, • understand the relationship between the exponential function f (x) = e x and the natural logarithm function f (x) = ln x.","PeriodicalId":130360,"journal":{"name":"Basic Analysis I","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125141949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Basic Analysis IPub Date : 2020-05-13DOI: 10.1201/9781315166254-20
J. Peterson
{"title":"Uniform Continuity","authors":"J. Peterson","doi":"10.1201/9781315166254-20","DOIUrl":"https://doi.org/10.1201/9781315166254-20","url":null,"abstract":"lim j→∞ xij = z. Let f : X → Y be a function with Y a metric space with metric ρ. We say that f is continuous at a point x0 ∈ X provided that for every > 0 there is a δ > 0 such that for every x ∈ X, if d(x, x0) < δ, then ρ(f(x), f(x0)) < . A function f : X → Y is said to be continuous provided that it is continuous at each point x ∈ X. A function f : X → Y is said to be uniformly continuous provided that for every > 0, there is a δ > 0 such that for every pair of points x and x′ in X, with d(x, x′) < δ, ρ(f(x), f(x′)) < . For a function f : X → Y to be uniformly continuous is stronger than being continuous.","PeriodicalId":130360,"journal":{"name":"Basic Analysis I","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129208595","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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