2b:null 2c:["$","title",null,{"children":"Intro Real Analysis: Continuity | Hans' workspace | Observable"}] 31:T5de,# Intro Real Analysis: Continuity Let ${tex`D \subset \mathbb{R}`}. We say a function ${tex`f: D \to \mathbb{R}`} is **continuous** at a point ${tex`x_0 \in D`} if for any ${tex`\epsilon > 0`}, there exists a ${tex`\delta > 0`} such that whenever ${tex`x \in D`} and ${tex`|x - x_0| < \delta`} we have ${tex`|f(x) - f(x_0)| < \epsilon`}. If ${tex`f`} is continuous at every ${tex`x_0 \in D`}, then we say ${tex`f`} is **continuous** on ${tex`D`}. Intuitively, for a continuous function, if we want to obtain function values that are arbitrarily close to ${tex`f(x_0)`} (i.e., in its ${tex`\epsilon`}-neighborhood), we just need to input domain values that are close enough to ${tex`x_0`} (i.e., in its ${tex`\delta`}-neighborhood). What makes the function continuous is that such a ${tex`\delta`}-neighborhood exists for any choice of ${tex`\epsilon`}, so there are no discontinuities in the function (or as is sometimes said, "you can draw the function without lifting your pen.") A function is **uniformly continuous** if for any ${tex`\epsilon > 0`}, there exists a ${tex`\delta > 0`} such that whenever ${tex`x,y \in D`} and ${tex`|x - y| < \delta`}, then ${tex`|f(x) - f(y)| < \epsilon`}. A key difference here is that there is no ${tex`x_0`} to which we are comparing. Rather, for a uniformly continuous function, we find a ${tex`\delta`} that works for any two points in ${tex`D`} that are within a ${tex`\delta`} distance of each other. This ${tex`\delta`} will not depend on ${tex`x_0`}.32:T4cc,## Example 1 Let's see our first, simple example: the linear function ${tex`f:\mathbb{R} \to \mathbb{R}`} defined as ${tex`f(x) = mx + b`} for some ${tex`m,b \in \mathbb{R}`} and all ${tex`x \in \mathbb{R}`}. We prove that ${tex`f`} is continuous. Thus, consider an arbitrary ${tex`x_0 \in \mathbb{R}`} and let ${tex`\epsilon > 0`}. Now, if ${tex`m = 0`}, then ${tex`f(x) = b`} for all ${tex`x \in \mathbb{R}`}, so any ${tex`\delta`} will work: no matter the value of ${tex`|x - x_0|`}, we will always have ${tex`|f(x) - f(x_0)| = 0 < \epsilon`}. Thus let's assume that ${tex`m \ne 0`}. Then take ${tex`\delta = \epsilon / |m|`}. Then when ${tex`x \in \mathbb{R}`} and ${tex`|x - x_0| < \delta`}, we have ${tex.block`|f(x) - f(x_0)| = |mx + b - (mx_0 + b)| = |m||x - x_0| < m (\epsilon / |m|) = \epsilon,`} implying that ${tex`f`} is continuous at ${tex`x_0`}. And since ${tex`x_0`} is arbitrary, we have that ${tex`f`} is continuous. Below, you can adjust the slope (and intercept) and choose your ${tex`\epsilon`}, which will determine the maximum ${tex`\delta`} that will work. Note that here the choice of ${tex`\delta`} does not depend on ${tex`x_0`}, which tells us that this function is actually uniformly continuous.33:T50d,## Example 2 Let's see an example that is just slightly more complicated. We consider the function ${tex`f(x) = x^2`} on ${tex`\mathbb{R}`}. Again this function is continuous. Here is one proof. Consider an arbitrary ${tex`x_0 \in \mathbb{R}`}, and let ${tex`\epsilon > 0`}. Pick ${tex`\delta > 0`} such that ${tex.block`\delta = \min \{1, \frac{\epsilon}{2|x_0| + 1}\}.`} Then whenever ${tex`x \in \mathbb{R}`} with ${tex`|x - x_0| < \delta`}, we have ${tex.block`|f(x) - f(x_0)| = |x^2 - x_0^2| = |x - x_0||x + x_0| = |x - x_0||x - x_0 + 2x_0|,`} and now applying the triangle inequality, ${tex.block`|x^2 - x_0^2| \le |x - x_0|(|x - x_0| + 2|x_0|) < \delta (\delta + 2|x_0|) < \delta (1 + 2 |x_0|) \le \epsilon,`} where we used that ${tex`\delta \le 1`}. This implies that ${tex`f`} is continuous at the arbitrary point ${tex`x_0`}. However, notice that the choice of ${tex`\delta`} depends on ${tex`x_0`}. This suggests that ${tex`f`} is not uniformly continuous, which can be proved true. Now, for your given choice of ${tex`\epsilon`}, as you change the value of ${tex`x_0`} you will notice that ${tex`\delta`} also changes. In particular, as ${tex`x_0`} moves further away from ${tex`0`} to where the the slope of ${tex`f(x)`} becomes steeper, the value of ${tex`\delta`} shrinks.34:[] 2f:{"id":"eec4be22a7677579","slug":null,"title":"Intro Real Analysis: Continuity","thumbnail":"25b1aeee40e9794bd5cb04790d4d9ef1fe76709517810cbe739570cf9b25ef5e","access_level":"public","create_time":"$D2025-12-19T03:09:55.222Z","update_time":"$D2026-10-01T01:44:03.921Z","view_time":null,"first_public_time":"$D2026-01-08T04:23:20.266Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"192cdf295ce772ed","login":"hans-workspace","name":"Hans' workspace","avatar_url":"https://avatars-next.observableusercontent.com/390dbc09f2362b985d98cc214aecf4f94e738ba4cfb05a3c5f435a3ce40a9eec","tier":"starter_2024","bio":""},"version":254,"publish_version":null,"role":null,"authored":false,"authors":[{"type":"individual","id":"7df739b561d8e445","login":"hans-elliott","name":"Hans 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${Math.round(delta*1000)/1000}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":25,"value":"Plot.plot({\n marks: [\n Plot.line(lineData, {x: \"x\", y: \"y\"}),\n\n // x0 point\n Plot.ruleX([x0], {stroke: \"#aaa\", strokeDasharray: \"8,8\"}),\n Plot.ruleY([m * x0 + b], {stroke: \"#aaa\", strokeDasharray: \"8,8\"}),\n\n Plot.dot([point], {\n x: \"x\",\n y: \"y\",\n r: 3,\n fill: \"black\"\n }),\n\n // f(x0) +- eps\n Plot.ruleY([m * x0 + b + epsilon], {stroke: \"red\"}),\n Plot.ruleY([m * x0 + b - epsilon], {stroke: \"red\"}),\n\n // x0 +- delta\n Plot.ruleX([x0 + delta], {stroke: \"lightgreen\"}),\n Plot.ruleX([x0 - delta], {stroke: \"lightgreen\"}),\n Plot.text(\n [{\n x: x0 + delta,\n y: -8,\n text: `delta = ${Math.round(delta*1000)/1000}`\n }],\n {\n x: \"x\",\n y: \"y\",\n text: \"text\",\n frameAnchor: \"top-left\",\n fontSize: 14,\n fontWeight: \"bold\"\n }\n )\n\n\n ],\n x: {domain: [-10, 10]},\n y: {domain: [-10, 10]},\n grid: false\n})\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":187,"value":"$33","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":208,"value":"viewof epsilon_2 = Inputs.range([0.1, 1], {steps: 0.1, value : 0.5, label: \"epsilon\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":210,"value":"viewof x0_2 = Inputs.range([-9, 9], {step: 0.1, label: \"Point of interest (x0)\"});","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":215,"value":"delta_2 = Math.min(1, epsilon_2 / (2 * Math.abs(x0_2) + 1))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":225,"value":"lineData_2 = d3.range(-9.01, 9.01, 0.1).map(x => ({\n x : x,\n y: x*x\n}))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":227,"value":"point_2 = ({ x: x0_2, y: x0_2 * x0_2})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"$undefined","database":"$undefined"},{"id":220,"value":"Given your choice of ${tex`\\epsilon`} and ${tex`x_0`}, we need ${tex`\\delta`} less than or equal to: ${Math.round(delta_2*1000)/1000}","pinned":false,"hidden":false,"mode":"md","author":"$undefined","summary":"$undefined","tool":"$undefined","providerMetadata":"$undefined","attachments":"$undefined","output":"","database":"$undefined"},{"id":218,"value":"Plot.plot({\n marks: [\n Plot.line(lineData_2, {x: \"x\", y: \"y\"}),\n\n // x0 point\n Plot.ruleX([x0_2], {stroke: \"#aaa\", strokeDasharray: \"8,8\"}),\n Plot.ruleY([x0_2 ** 2], {stroke: \"#aaa\", strokeDasharray: \"8,8\"}),\n\n Plot.dot([point_2], {\n x: \"x\",\n y: \"y\",\n r: 3,\n fill: \"black\"\n }),\n\n // f(x0) +- eps\n Plot.ruleY([x0_2**2 + epsilon_2], {stroke: \"red\"}),\n 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