// Numbas version: exam_results_page_options
{"name": "True/false statements about continuity at a point", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"del": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('$\\\\epsilon$','$\\\\chi$','$\\\\rho$','$\\\\omega$')", "name": "del", "description": ""}, "ep": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random('$\\\\alpha$','$\\\\beta$','$\\\\gamma$','$\\\\delta$')", "name": "ep", "description": ""}}, "ungrouped_variables": ["del", "ep"], "name": "True/false statements about continuity at a point", "functions": {}, "parts": [{"showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "prompt": "<p>Choose the correct definitions of continuity at $x_0$ from the following:</p>\n<p></p>\n<p>[[0]]</p>", "unitTests": [], "sortAnswers": false, "scripts": {}, "gaps": [{"displayType": "checkbox", "showCorrectAnswer": true, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "matrix": [2, 0, 0, 0, 2, 0, 2], "choices": ["<p>For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{ep} \\gt 0$ , there exists a $\\var{del} \\gt 0$ such that $\\left |x-x_0\\right | \\lt \\var{ep}$ whenever $|f(x)-f(x_0)| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>There exists $\\var{ep} \\gt 0$ , such that for every $\\var{del} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>For every $\\var{del} \\gt 0$ , there exists a $\\var{ep} \\gt 0$ such that $\\left |f(x)-f(x_0)\\right | \\lt \\var{del}$ whenever $|x-x_0| \\lt \\var{ep}$ and $x \\in I$.</p>", "<p>There exists $\\var{del} \\gt 0$ , such that for every $\\var{ep} \\gt 0$, $\\left |f(x)-f(x_0)\\right | \\lt \\var{ep}$ whenever $|x-x_0| \\lt \\var{del}$ and $x \\in I$.</p>", "<p>$\\lim_{x \\to x_0}f(x)=f(x_0)$.</p>"], "unitTests": [], "distractors": ["", "", "", "", "", "", ""], "variableReplacements": [], "type": "m_n_2", "maxAnswers": 0, "shuffleChoices": true, "showFeedbackIcon": true, "scripts": {}, "maxMarks": 0, "minAnswers": 0, "minMarks": 0, "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 1, "marks": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "marks": 0, "showFeedbackIcon": true}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>Let $\\mathbb{R}$ be the set of real numbers.</p>\n<p>Let $x_0$ be a point in the open interval $I \\subset \\mathbb{R}$ and let $f:I &nbsp;\\rightarrow \\mathbb{R}$ be a function.</p>\n<p>What does it mean to say that $f$ is continuous at $x_0$?&nbsp;</p>\n<p></p>", "tags": ["checked2015", "continuity", "continuity at a point", "continuous functions", "definition of continuity"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": ""}, "advice": "", "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}, {"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}