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En raison de limitations techniques, la typographie souhaitable du titre, «
Exercice : ContinuitéFonctions d'une variable réelle/Exercices/Continuité », n'a pu être restituée correctement ci-dessus.
Soient
−
∞
≤
a
<
b
≤
+
∞
{\displaystyle -\infty \leq a<b\leq +\infty }
f
:
]
a
,
b
[
→
R
{\displaystyle f:\left]a,b\right[\to \mathbb {R} }
On suppose que
f
{\displaystyle f}
a
{\displaystyle a}
b
{\displaystyle b}
l
:=
lim
a
f
et
L
:=
lim
b
f
{\displaystyle l:=\lim _{a}f\quad {\text{et}}\quad L:=\lim _{b}f}
.
Montrer que
f
{\displaystyle f}
l
{\displaystyle l}
L
{\displaystyle L}
Montrer que si
l
{\displaystyle l}
L
{\displaystyle L}
f
{\displaystyle f}
Solution
Soit un réel
M
1
>
max
(
|
l
|
,
|
L
|
)
{\displaystyle M_{1}>\max(|l|,|L|)}
A
{\displaystyle A}
B
{\displaystyle B}
a
<
A
≤
B
<
b
et
∀
x
∈
]
a
,
A
[
∪
]
B
,
b
[
|
f
(
x
)
|
<
M
1
{\displaystyle a<A\leq B<b\quad {\text{et}}\quad \forall x\in \left]a,A\right[\cup \left]B,b\right[\quad |f(x)|<M_{1}}
Posons
M
2
=
sup
|
f
|
(
[
A
,
B
]
)
{\displaystyle M_{2}=\sup |f|\left(\left[A,B\right]\right)}
∀
x
∈
]
a
,
b
[
|
f
(
x
)
|
≤
max
(
M
1
,
M
2
)
{\displaystyle \forall x\in \left]a,b\right[\quad |f(x)|\leq \max(M_{1},M_{2})}
On suppose que
l
=
L
{\displaystyle l=L}
1) Montrer que si
f
{\displaystyle f}
L
=
+
∞
{\displaystyle L=+\infty }
f
{\displaystyle f}
Conseil : Rien ne vaut un bon schéma. Il faut alors utiliser la définition de la limite et…2) En déduire que (sans cette dernière hypothèse)
f
{\displaystyle f}
Solution
1) Soit
c
∈
]
a
,
b
[
{\displaystyle c\in \left]a,b\right[}
f
(
c
)
<
L
{\displaystyle f(c)<L}
A
{\displaystyle A}
B
{\displaystyle B}
∀
x
∈
]
a
,
A
[
∪
]
B
,
b
[
f
(
x
)
>
f
(
c
)
{\displaystyle \forall x\in \left]a,A\right[\cup \left]B,b\right[\quad f(x)>f(c)}
A
≤
c
≤
B
{\displaystyle A\leq c\leq B}
D'après le théorème des bornes, la restriction de
f
{\displaystyle f}
[
A
,
B
]
{\displaystyle \left[A,B\right]}
d
{\displaystyle d}
f
(
d
)
≤
f
(
c
)
{\displaystyle f(d)\leq f(c)}
∀
x
∈
]
a
,
b
[
f
(
x
)
≥
f
(
d
)
{\displaystyle \forall x\in \left]a,b\right[\quad f(x)\geq f(d)}
2) Si
f
{\displaystyle f}
Supposons maintenant que
f
{\displaystyle f}
c
∈
]
a
,
b
[
{\displaystyle c\in \left]a,b\right[}
L
{\displaystyle L}
Si
f
(
c
)
<
L
{\displaystyle f(c)<L}
f
{\displaystyle f}
f
(
c
)
>
L
{\displaystyle f(c)>L}
f
{\displaystyle f}
−
f
{\displaystyle -f}
Pour une généralisation des exercices 2 et 3, voir Espaces vectoriels normés/Exercices/Dimension finie#Exercice 3-3 : extrema d'une fonction continue (niveau 15).
On pose :
arctan
(
−
∞
)
=
−
π
/
2
,
arctan
(
+
∞
)
=
π
/
2
,
a
′
=
arctan
(
a
)
,
b
′
=
arctan
(
b
)
{\displaystyle \arctan(-\infty )=-\pi /2,~\arctan(+\infty )=\pi /2,~a'=\arctan(a),~b'=\arctan(b)}
1) Redémontrer le résultat de l'exercice 2 en prolongeant par continuité la fonction
f
∘
tan
:
]
a
′
,
b
′
[
→
R
{\displaystyle f\circ \tan :\left]a',b'\right[\to \mathbb {R} }
2) Redémontrer le résultat de l'exercice 1 en prolongeant par continuité la fonction
arctan
∘
f
∘
tan
:
]
a
′
,
b
′
[
→
R
{\displaystyle \arctan \circ f\circ \tan :\left]a',b'\right[\to \mathbb {R} }
3) Redémontrer les résultats de l'exercice 3 à l'aide du même prolongement de
arctan
∘
f
∘
tan
{\displaystyle \arctan \circ f\circ \tan }
Solution
1) Si
l
{\displaystyle l}
L
{\displaystyle L}
f
∘
tan
{\displaystyle f\circ \tan }
g
:
[
a
′
,
b
′
]
→
R
{\displaystyle g:\left[a',b'\right]\to \mathbb {R} }
g
(
a
′
)
=
l
{\displaystyle g(a')=l}
g
(
b
′
)
=
L
{\displaystyle g(b')=L}
g
{\displaystyle g}
f
(
]
a
,
b
[
)
=
g
(
]
a
′
,
b
′
[
)
{\displaystyle f\left(\left]a,b\right[\right)=g\left(\left]a',b'\right[\right)}
2) Même si
l
{\displaystyle l}
L
{\displaystyle L}
arctan
∘
f
∘
tan
{\displaystyle \arctan \circ f\circ \tan }
h
{\displaystyle h}
[
a
′
,
b
′
]
{\displaystyle \left[a',b'\right]}
y
{\displaystyle y}
l
{\displaystyle l}
L
{\displaystyle L}
arctan
y
{\displaystyle \arctan y}
arctan
l
=
h
(
a
′
)
{\displaystyle \arctan l=h(a')}
arctan
L
=
h
(
b
′
)
{\displaystyle \arctan L=h(b')}
∃
c
′
∈
]
a
′
,
b
′
[
arctan
y
=
h
(
c
′
)
{\displaystyle \exists c'\in \left]a',b'\right[\quad \arctan y=h(c')}
d'où, en posant
c
=
tan
(
c
′
)
{\displaystyle c=\tan(c')}
c
∈
]
a
,
b
[
et
y
=
f
(
c
)
{\displaystyle c\in \left]a,b\right[{\text{ et }}y=f(c)}
3) La fonction
h
{\displaystyle h}
l
=
L
{\displaystyle l=L}
h
(
a
′
)
=
h
(
b
′
)
{\displaystyle h(a')=h(b')}
inf
f
<
l
=
L
{\displaystyle \inf f<l=L}
min
h
<
h
(
a
′
)
=
h
(
b
′
)
{\displaystyle \min h<h(a')=h(b')}
h
{\displaystyle h}
c
′
∈
]
a
′
,
b
′
[
{\displaystyle c'\in \left]a',b'\right[}
f
{\displaystyle f}
c
:=
tan
c
′
{\displaystyle c:=\tan c'}
sup
f
>
l
=
L
{\displaystyle \sup f>l=L}
h
{\displaystyle h}
c
′
∈
]
a
′
,
b
′
[
{\displaystyle c'\in \left]a',b'\right[}
f
{\displaystyle f}
c
:=
tan
c
′
{\displaystyle c:=\tan c'}
h
{\displaystyle h}
f
{\displaystyle f}
Sur
]
0
,
+
∞
[
{\displaystyle \left]0,+\infty \right[}
f
{\displaystyle f}
g
:
x
↦
f
(
x
)
/
x
{\displaystyle g:x\mapsto f(x)/x}
Montrer que
f
{\displaystyle f}
Montrer que si
f
{\displaystyle f}
Donner un exemple de telle fonction.
Solution
Si
0
<
x
<
y
{\displaystyle 0<x<y}
f
(
x
)
≤
f
(
y
)
{\displaystyle f(x)\leq f(y)}
f
(
x
)
/
x
≥
f
(
y
)
/
y
{\displaystyle f(x)/x\geq f(y)/y}
f
(
x
)
≤
f
(
y
)
≤
f
(
x
)
y
/
x
{\displaystyle f(x)\leq f(y)\leq f(x)y/x}
y
→
x
+
{\displaystyle y\to x^{+}}
f
(
y
)
→
f
(
x
)
{\displaystyle f(y)\to f(x)}
0
<
y
<
x
{\displaystyle 0<y<x}
f
(
y
)
≤
f
(
x
)
{\displaystyle f(y)\leq f(x)}
f
(
y
)
/
y
≥
f
(
x
)
/
x
{\displaystyle f(y)/y\geq f(x)/x}
f
(
x
)
y
/
x
≤
f
(
y
)
≤
f
(
x
)
{\displaystyle f(x)y/x\leq f(y)\leq f(x)}
y
→
x
+
{\displaystyle y\to x^{+}}
f
(
y
)
→
f
(
x
)
{\displaystyle f(y)\to f(x)}
Remarquons d'abord que pour tout
x
>
0
{\displaystyle x>0}
f
(
x
)
≥
0
{\displaystyle f(x)\geq 0}
∀
y
∈
]
0
,
x
[
f
(
x
)
y
/
x
≤
f
(
x
)
{\displaystyle \forall y\in \left]0,x\right[\quad f(x)y/x\leq f(x)}
lim
y
→
0
+
f
(
x
)
y
/
x
=
0
{\displaystyle \lim _{y\to 0^{+}}f(x)y/x=0}
f
{\displaystyle f}
x
{\displaystyle x}
∀
y
∈
]
0
,
x
[
0
≤
f
(
y
)
≤
f
(
x
)
=
0
{\displaystyle \forall y\in \left]0,x\right[\quad 0\leq f(y)\leq f(x)=0}
f
(
y
)
=
0
{\displaystyle f(y)=0}
∀
y
∈
]
x
,
+
∞
[
0
≤
f
(
y
)
≤
f
(
x
)
y
/
x
=
0
{\displaystyle \forall y\in \left]x,+\infty \right[\quad 0\leq f(y)\leq f(x)y/x=0}
f
(
y
)
=
0
{\displaystyle f(y)=0}
On peut choisir par exemple
f
{\displaystyle f}
g
{\displaystyle g}
f
(
x
)
=
c
{\displaystyle f(x)=c}
f
(
x
)
=
c
x
{\displaystyle f(x)=cx}
c
>
0
{\displaystyle c>0}
f
{\displaystyle f}
g
{\displaystyle g}
f
(
x
)
=
a
x
+
b
{\displaystyle f(x)=ax+b}
a
,
b
>
0
{\displaystyle a,b>0}
Montrer que
lim
x
→
+
∞
f
(
x
)
x
=
+
∞
{\displaystyle \lim _{x\to +\infty }{\frac {f(x)}{x}}=+\infty }
∀
a
∈
R
lim
x
→
+
∞
f
(
x
)
−
a
x
=
+
∞
{\displaystyle \forall a\in \mathbb {R} \quad \lim _{x\to +\infty }f(x)-ax=+\infty }
Solution
⇒
{\displaystyle \Rightarrow }
lim
x
→
+
∞
f
(
x
)
x
=
+
∞
{\displaystyle \lim _{x\to +\infty }{\frac {f(x)}{x}}=+\infty }
a
,
b
∈
R
{\displaystyle a,b\in \mathbb {R} }
M
=
a
+
max
(
b
,
0
)
{\displaystyle M=a+\max(b,0)}
A
≥
1
{\displaystyle A\geq 1}
x
>
A
{\displaystyle x>A}
f
(
x
)
x
>
M
{\displaystyle {\frac {f(x)}{x}}>M}
f
(
x
)
−
a
x
>
max
(
b
,
0
)
x
≥
b
×
1
{\displaystyle f(x)-ax>\max(b,0)x\geq b\times 1}
lim
x
→
+
∞
f
(
x
)
−
a
x
=
+
∞
{\displaystyle \lim _{x\to +\infty }f(x)-ax=+\infty }
⇐
{\displaystyle \Leftarrow }
∀
a
∈
R
lim
x
→
+
∞
f
(
x
)
−
a
x
=
+
∞
{\displaystyle \forall a\in \mathbb {R} \quad \lim _{x\to +\infty }f(x)-ax=+\infty }
M
∈
R
{\displaystyle M\in \mathbb {R} }
A
≥
0
{\displaystyle A\geq 0}
x
>
A
{\displaystyle x>A}
f
(
x
)
−
M
x
>
0
{\displaystyle f(x)-Mx>0}
f
(
x
)
x
>
M
{\displaystyle {\frac {f(x)}{x}}>M}
lim
x
→
+
∞
f
(
x
)
x
=
+
∞
{\displaystyle \lim _{x\to +\infty }{\frac {f(x)}{x}}=+\infty }
![{\displaystyle f:\left]a,b\right[\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee5b56aa82e56ebac7378f0a91ad1851e3810bd9)
![{\displaystyle a<A\leq B<b\quad {\text{et}}\quad \forall x\in \left]a,A\right]\quad f(x)\leq y\quad {\text{et}}\quad \forall x\in \left[B,b\right[\quad f(x)\geq y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0bc899c51fe28b421bfbe32557f7a81fcbb9b2)
![{\displaystyle \exists x\in \left[A,B\right]\quad y=f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c42f29fe6e82257e3dc770d180daa476af18e808)
![{\displaystyle a<A\leq B<b\quad {\text{et}}\quad \forall x\in \left]a,A\right[\cup \left]B,b\right[\quad |f(x)|<M_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3330d47a024a3485206213440ebbe6049981dbde)
![{\displaystyle M_{2}=\sup |f|\left(\left[A,B\right]\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cc9272bcb514b3364eda440e3dbe7ee843b5412)
![{\displaystyle \forall x\in \left]a,b\right[\quad |f(x)|\leq \max(M_{1},M_{2})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d2a19ee1d7addf714ef6cf99197a50343060fb)
![{\displaystyle c\in \left]a,b\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/949411efe3a008e113a12114de62ab2142344048)
![{\displaystyle \forall x\in \left]a,A\right[\cup \left]B,b\right[\quad f(x)>f(c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb18470694b6536c8a7cb75c90715bf336ae897)
![{\displaystyle \left[A,B\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8dc0b2c2be2f061523a5eac3de16ba3761ed019)
![{\displaystyle \forall x\in \left]a,b\right[\quad f(x)\geq f(d)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69458a169ae003d0072c2a7252a3761ac9a8d62d)
![{\displaystyle f\circ \tan :\left]a',b'\right[\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3dafd91186d81e698080f6770e5125115a6981b4)
![{\displaystyle \arctan \circ f\circ \tan :\left]a',b'\right[\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/87b734341be21332fcdf948e4edf2ebafacb99d0)
![{\displaystyle g:\left[a',b'\right]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/79fe43b1b1a9f75e269a62f91d38f1ba7a597846)
![{\displaystyle f\left(\left]a,b\right[\right)=g\left(\left]a',b'\right[\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/774d851d7261f911722f0358452a615977850d04)
![{\displaystyle \left[a',b'\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb25b6f10cea8a9a2c7bd749f88036584e4a789b)
![{\displaystyle \exists c'\in \left]a',b'\right[\quad \arctan y=h(c')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0174142267fc0d6717329827290050f94bf1bf5)
![{\displaystyle c\in \left]a,b\right[{\text{ et }}y=f(c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4596ab915c241712f3ab1e1a0486abbbb7f022a8)
![{\displaystyle c'\in \left]a',b'\right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08006a5bac97ec209530defad271a6096a1e4517)
![{\displaystyle \left]0,+\infty \right[}](https://wikimedia.org/api/rest_v1/media/math/render/svg/010ebc9c2d0f3af5ce1c0594c44823f107968561)
![{\displaystyle \forall y\in \left]0,x\right[\quad f(x)y/x\leq f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7918b5891ed3e7e4a5d2640d3f1888bdea8a714)
![{\displaystyle \forall y\in \left]0,x\right[\quad 0\leq f(y)\leq f(x)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0321eb0c149797ee9d47304a0666091018d22b8b)
![{\displaystyle \forall y\in \left]x,+\infty \right[\quad 0\leq f(y)\leq f(x)y/x=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27b96665b5d895af232a70ce72e82913a28cb6a8)
