Source / pinned edition
Continuity
Source of package openstax.calculus_limits · OpenStax Calculus, ch. 2 Limits, module m53489
- work
- urn:openstax:clir:calculus-limits#M53489
- number
- OpenStax Calculus, ch. 2 Limits, module m53489
- kind
- urn:openstax:clir:calculus-limits#treatise
m53489/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-limits#M53489_MAIN
- language
- en
- officiality
- official
- materialisation
- PINNED_OFFICIAL_BYTES
- life cycle
- adopted: 2026-07-15
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53489/index.cnxml | application/xml; charset=utf-8 | sha256:96e89d1cd16c4be5a87a3d68d19f2552204ffc97f6316433ef35689187c5bf7f | 2026-09-12T00:00:00Z |
Fragments — 2
module/m53489#fs-id1170571048312 · defn · referenced by: 1
en · official · sha256:3292e4251fecd670028ac300a28054b5163a3844b6a6e39b0b722ad65aab7661
<note id="fs-id1170571048312">
<title>Definition</title>
<para id="fs-id1170573381317">A function <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> is <term id="term-00001">continuous at a point</term> <emphasis effect="italics">a</emphasis> if and only if the following three conditions are satisfied:</para>
<list id="fs-id1170571094915" list-type="enumerated" number-style="lower-roman">
<item><m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>a</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> is defined</item>
<item><m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi><m:mo stretchy="false">→</m:mo><m:mi>a</m:mi></m:mrow></m:munder><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> exists</item>
<item><m:math><m:mrow><m:munder><m:mrow><m:mtext>lim</m:mtext></m:mrow><m:mrow><m:mi>x</m:mi><m:mo stretchy="false">→</m:mo><m:mi>a</m:mi></m:mrow></m:munder><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>a</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math></item>
</list>
<para id="fs-id1170573370662">A function is <term id="term-00002">discontinuous at a point</term> <emphasis effect="italics">a</emphasis> if it fails to be continuous at <emphasis effect="italics">a</emphasis>.</para>
</note>Referencing norms: JustifyContinuity11
module/m53489#fs-id1170573418920 · section · referenced by: 1
en · official · sha256:14f7acb9225c4cbde4680d8a8e3289776376ca5bd0eb419e13f9e6ad47de6719
<note id="fs-id1170573418920" class="theorem">
<title>Continuity of Trigonometric Functions</title>
<para id="fs-id1170573512311">Trigonometric functions are continuous over their entire domains.</para>
</note>Referencing norms: JustifyContinuity12
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