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Newsgroups: fr.sci.maths
Date: Mon, 11 Jul 2022 12:48:01 -0700 (PDT)
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Subject: =?UTF-8?Q?Re=3A_Abscisses_de_discontinuit=C3=A9?=
From: did <didier.oslo@hotmail.com>
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Oui, j'avais remarqu=C3=A9 que la fonction est impaire, sauf aux points de =
discontinuit=C3=A9.=20
J'aurais d=C3=BB le pr=C3=A9ciser. Merci pour la d=C3=A9monstration.=20

Pour les abscisses de discontinuit=C3=A9 (et la hauteur des sauts), je ne v=
ois pas comment=20
m'y prendre.=20

On Monday, 11 July 2022 at 21:23:45 UTC+2, Olivier Miakinen wrote:
> Le 11/07/2022 20:41, Olivier Miakinen a =C3=A9crit :=20
> > Le 11/07/2022 20:31, did a =C3=A9crit :=20
> >> J'ai ajout=C3=A9 le PS trop vite sans v=C3=A9rifier.=20
> >> f n'est pas impaire,=20
> >=20
> > En effet. f(1) =3D f(1/2) =3D -6 alors que f(-1) =3D 5 et f(-1/2) =3D -=
1=20
> >=20
> >> c'est une autre fonction dans=20
> >> laquelle elle apparait. En fait, la fonction qui=20
> >> m'int=C3=A9resse vraiment est=20
> >> F(x) =3D 1/2 + [ x + 1/2 ] + [ x - 2 * pi * [ x + 1/2 ] ],=20
> >> qui semble =C3=AAtre impaire d'apr=C3=A8s son graphe,=20
> >> mais cela reste =C3=A0 d=C3=A9montrer.
> Alors.=20
>=20
> Cette fonction n'est *pas* impaire, parce que par exemple f(0) =3D 1/2 =
=E2=89=A0 0.=20
> En revanche je peux montrer que F(=E2=88=92x) =3D =E2=88=92F(x) partout /=
sauf/ aux points de=20
> discontinuit=C3=A9 !=20
>=20
> D=C3=A9j=C3=A0, pardon pour l'anglais, je vais noter floor(x) =3D =E2=8C=
=8Ax=E2=8C=8B et ceil(x) =3D =E2=8C=88x=E2=8C=89,=20
> =C3=A7a me semble plus facile =C3=A0 =C3=A9crire et m=C3=AAme =C3=A0 lire=
..=20
>=20
>=20
> 1. Quelques remarques pr=C3=A9liminaires=20
>=20
> Aux points de discontinuit=C3=A9, on a ceil(x) =3D floor(x) =3D x. Ces po=
ints ne vont=20
> pas nous int=C3=A9resser.=20
>=20
> En dehors d'un point de discontinuit=C3=A9, on a :=20
> (I) ceil(x) =3D floor(x) + 1 =3D floor(x + 1).=20
>=20
> Par ailleurs, pour tout x, on a :=20
> (II) floor(x) =3D =E2=88=92 ceil(=E2=88=92x) et ceil(x) =3D =E2=88=92 flo=
or(=E2=88=92x).=20
>=20
>=20
> 2. Allons-y pour les calculs=20
>=20
> On part de :=20
> F(x) =3D 1/2 + floor(x + 1/2) + floor(x =E2=88=92 2 pi floor(x + 1/2))=20
>=20
> En dehors de tout point de discontinuit=C3=A9, on doit avoir :=20
> F(=E2=88=92x) =3D 1/2 + floor(=E2=88=92x + 1/2) + floor(=E2=88=92x =E2=88=
=92 2 pi floor(=E2=88=92x + 1/2))=20
> F(=E2=88=92x) =3D 1/2 =E2=88=92 ceil(x =E2=88=92 1/2) + floor(=E2=88=92x =
+ 2 pi ceil(x =E2=88=92 1/2)) (par II)=20
> F(=E2=88=92x) =3D 1/2 =E2=88=92 floor(x =E2=88=92 1/2 + 1) + floor(=E2=88=
=92x + 2 pi floor(x =E2=88=92 1/2 + 1)) (par I)=20
> F(=E2=88=92x) =3D 1/2 =E2=88=92 floor(x + 1/2) + floor(=E2=88=92x + 2 pi =
floor(x + 1/2))=20
> F(=E2=88=92x) =3D 1/2 =E2=88=92 floor(x + 1/2) =E2=88=92 ceil(x =E2=88=92=
 2 pi floor(x + 1/2)) (par II)=20
> F(=E2=88=92x) =3D 1/2 =E2=88=92 floor(x + 1/2) =E2=88=92 floor(x =E2=88=
=92 2 pi floor(x + 1/2)) =E2=88=92 1 (par I)=20
> F(=E2=88=92x) =3D =E2=88=921/2 =E2=88=92 floor(x + 1/2) =E2=88=92 floor(x=
 =E2=88=92 2 pi floor(x + 1/2))=20
> F(=E2=88=92x) =3D =E2=88=92 F(x), CQFD=20
>=20
>=20
> --=20
> Olivier Miakinen