{"id":162,"date":"2021-07-30T17:23:14","date_gmt":"2021-07-30T17:23:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=162"},"modified":"2022-10-29T01:17:59","modified_gmt":"2022-10-29T01:17:59","slug":"summary-of-limits-and-continuity","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-limits-and-continuity\/","title":{"raw":"Summary of Limits and Continuity","rendered":"Summary of Limits and Continuity"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<ul>\r\n \t<li>To study limits and continuity for functions of two variables, we use a [latex]\\delta[\/latex]\u00a0disk centered around a given point.<\/li>\r\n \t<li>A function of several variables has a limit if for any point in a\u00a0[latex]\\delta[\/latex]\u00a0ball centered at a point [latex]P[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value).<\/li>\r\n \t<li>The limit laws established for a function of one variable have natural extensions to functions of more than one variable.<\/li>\r\n \t<li>A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>boundary point<\/dt>\r\n \t<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>closed set<\/dt>\r\n \t<dd>a set [latex]S[\/latex]<strong>\u00a0<\/strong>that contains all its boundary points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>connected set<\/dt>\r\n \t<dd>an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>interior point<\/dt>\r\n \t<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>open set<\/dt>\r\n \t<dd>a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>region<\/dt>\r\n \t<dd>an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt><strong>[latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\r\n \t<dd>all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt><strong>[latex]\\delta[\/latex] d<\/strong>isk<\/dt>\r\n \t<dd>an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<ul>\n<li>To study limits and continuity for functions of two variables, we use a [latex]\\delta[\/latex]\u00a0disk centered around a given point.<\/li>\n<li>A function of several variables has a limit if for any point in a\u00a0[latex]\\delta[\/latex]\u00a0ball centered at a point [latex]P[\/latex],\u00a0the value of the function at that point is arbitrarily close to a fixed value (the limit value).<\/li>\n<li>The limit laws established for a function of one variable have natural extensions to functions of more than one variable.<\/li>\n<li>A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.<\/li>\n<\/ul>\n<\/div>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>boundary point<\/dt>\n<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of [latex]R[\/latex]\u00a0is a boundary point if every [latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contains points both inside and outside [latex]R[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>closed set<\/dt>\n<dd>a set [latex]S[\/latex]<strong>\u00a0<\/strong>that contains all its boundary points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>connected set<\/dt>\n<dd>an open set [latex]S[\/latex]\u00a0that cannot be represented as the union of two or more disjoint, nonempty open subsets<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>interior point<\/dt>\n<dd>a point\u00a0[latex]P_{0}[\/latex]\u00a0of\u00a0[latex]R[\/latex]\u00a0is a boundary point if there is a\u00a0[latex]\\delta[\/latex]\u00a0disk centered around [latex]P_{0}[\/latex]\u00a0contained completely in\u00a0[latex]R[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>open set<\/dt>\n<dd>a set\u00a0[latex]S[\/latex]\u00a0that contains none of its boundary points<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>region<\/dt>\n<dd>an open, connected, nonempty subset of [latex]\\mathbb{R}^{2}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>[latex]\\delta[\/latex]<\/strong>\u00a0ball<\/dt>\n<dd>all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than\u00a0[latex]\\delta[\/latex]\u00a0from [latex](x_{0},y_{0},z_{0})[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt><strong>[latex]\\delta[\/latex] d<\/strong>isk<\/dt>\n<dd>an open disk of radius [latex]\\delta[\/latex]\u00a0centered at point\u00a0[latex](a,b)[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-162\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":10,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-162","chapter","type-chapter","status-publish","hentry"],"part":22,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/162","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/17533"}],"version-history":[{"count":13,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions"}],"predecessor-version":[{"id":3764,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/162\/revisions\/3764"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/162\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=162"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=162"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=162"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=162"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}