{"id":235638,"date":"2022-06-13T17:15:00","date_gmt":"2022-06-13T15:15:00","guid":{"rendered":"https:\/\/sherpas.com\/blog\/?p=235638"},"modified":"2025-09-29T11:53:39","modified_gmt":"2025-09-29T09:53:39","slug":"montrer-quune-fonction-est-lipschitzienne","status":"publish","type":"post","link":"https:\/\/sherpas.com\/blog\/montrer-quune-fonction-est-lipschitzienne\/","title":{"rendered":"Comment montrer qu’une fonction est lipschitzienne ?"},"content":{"rendered":"\n

Vous cherchez \u00e0 montrer qu’une fonction est lipschitzienne<\/strong> ? Gr\u00e2ce \u00e0 ce cours d\u00e9di\u00e9 \u00e0 la notion de fonction lipschitzienne<\/strong>, ma\u00eetriser ce chapitre sur le bout des doigts gr\u00e2ce \u00e0 des m\u00e9thodologies compl\u00e8tes et adapt\u00e9es !<\/p>\n

Avant ton prochain DS, assure-toi que la notion de fonction lipschitzienne est bien ancr\u00e9e avec nos cours de soutien en math\u00e9matiques<\/strong><\/a>, et aborde ton examen en toute confiance ! \ud83d\udcda<\/span><\/p>\n\n\n\n

M\u00e9thode : Montrer qu’une fonction est lipschitzienne<\/h2>\n\n\n\n

Conseils m\u00e9thodologiques<\/h3>\n\n\n\n

Pour montrer qu’une fonction est lipschitzienne, on peut utiliser l’in\u00e9galit\u00e9 des accroissements finis.<\/p>\n\n\n\n

Application de la m\u00e9thode : <\/h3>\n\n\n\n

<\/p>\nMontrons que \"\arctan\" est lipschitzienne sur \"\mathbb{R}\".
\nLa fonction \"\arctan\" est d\u00e9rivable sur \"\mathbb{R}\" et, pour tout \"x\in\mathbb{R}\", \"|\arctan'(x)|=\dfrac{1}{1+x^2}\leq 1\".
\nDonc, par l’in\u00e9galit\u00e9 des accroissements finis, \"\forall(x,y)\in\R^2,\quad |\arctan(y)-\arctan(x)|\leq |y-x|.\"\n\n\n\n

\n
\n
\"livre<\/figure>\n<\/div>\n\n\n\n
<\/div>\n\n\n\n
\n
<\/div>\n
\n

Cet article est extrait de l’ouvrage Maths MPSI-MP2I. Tout-en-un : cours, m\u00e9thodes, entra\u00eenement et corrig\u00e9s <\/em>(\u00e9ditions Vuibert, juin 2021) \u00e9crit par E. Thomas, S. Bellec, G. Boutard. ISBN n\u00b09782311408720<\/em><\/p>\n\n <\/div>\n <\/section>\n<\/div>\n<\/div>\n\n\n

\n \n
\n \n
\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n <\/div>\n \n
\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n
\n \n\n
<\/div>\n <\/div>\n <\/div>\n<\/div>\n \n\n
\n 4.1\/5 - (12 votes) <\/div>\n <\/div>\n","protected":false},"excerpt":{"rendered":"

Vous cherchez \u00e0 montrer qu’une fonction est lipschitzienne ? Gr\u00e2ce \u00e0 ce cours d\u00e9di\u00e9 \u00e0 la notion de (…)<\/p>\n","protected":false},"author":158,"featured_media":165125,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":true,"footnotes":""},"category":[803,810],"tag":[78,345],"class_list":["post-235638","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-apprendre-matiere","category-maths","tag-prepa","tag-prepa-scientifique"],"acf":[],"_links":{"self":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/235638","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/users\/158"}],"replies":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/comments?post=235638"}],"version-history":[{"count":0,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/posts\/235638\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media\/165125"}],"wp:attachment":[{"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/media?parent=235638"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/category?post=235638"},{"taxonomy":"tag","embeddable":true,"href":"https:\/\/sherpas.com\/blog\/wp-json\/wp\/v2\/tag?post=235638"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}