Graphum cuiusdam functionis differentiabilis continuaeque in intervallo [-1,1.5] definitae:
f
:
[
−
1
,
1.5
]
→
[
−
1
,
1.5
]
x
↦
(
4
x
3
−
6
x
2
+
1
)
x
+
1
3
−
x
{\displaystyle {\begin{aligned}&\scriptstyle f\colon [-1,1.5]\to [-1,1.5]\\&\textstyle x\mapsto {\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}
Continuitas in topologia est functio realis dominii
f
:
I
→
R
{\displaystyle f:I\to \mathbb {R} }
intervallo reali
I
⊆
R
{\displaystyle I\subseteq \mathbb {R} }
graphum functionis
f
{\displaystyle f}
mathematicae quae de functionibus continuis tractat est analysis ; pars quae de rebus discretis tractat est mathematica discreta , in qua sunt calculus coniunctionibus vel "ars combinatoria," arithmetica integrorum , et probabilitas .
Sunt duae definitiones continuitatis quae aequae sunt:
Regula Epsilon-Delta [1]
f
:
D
→
R
{\displaystyle f\colon D\to \mathbb {R} }
continua in
x
0
∈
D
{\displaystyle x_{0}\in D}
est, si
ε
>
0
{\displaystyle \varepsilon >0}
δ
>
0
{\displaystyle \delta >0}
x
∈
D
{\displaystyle x\in D}
|
x
−
x
0
|
<
δ
{\displaystyle |x-x_{0}|<\delta }
|
f
(
x
)
−
f
(
x
0
)
|
<
ε
{\displaystyle |f(x)-f(x_{0})|<\varepsilon }
Regula sequentiarum [2]
f
:
D
→
R
{\displaystyle f\colon D\to \mathbb {R} }
continua in
x
0
∈
D
{\displaystyle x_{0}\in D}
, si, cum quaelibet sequentia
(
x
k
)
k
∈
N
{\displaystyle (x_{k})_{k\in \mathbb {N} }}
(
x
k
∈
D
)
{\displaystyle (x_{k}\in D)}
x
0
{\displaystyle x_{0}}
f
(
x
k
)
{\displaystyle f(x_{k})}
f
(
x
0
)
{\displaystyle f(x_{0})}
Functio appellatur continua in
D
{\displaystyle D}
, si est continua in locis omnibus dominii.
Si est
x
0
∈
D
{\displaystyle x_{0}\in D}
discontinua appellatur.
Functiones
sin
:
R
→
R
,
x
↦
sin
x
{\displaystyle \sin \colon \mathbb {R} \to \mathbb {R} ,\;x\mapsto \sin x}
cos
:
R
→
R
,
x
↦
cos
x
{\displaystyle \cos \colon \mathbb {R} \to \mathbb {R} ,\;x\mapsto \cos x}
R
{\displaystyle \mathbb {R} }
Functio signi
sgn
(
x
)
=
{
1
,
x
>
0
,
0
,
x
=
0
,
−
1
,
x
<
0
,
{\displaystyle \ \operatorname {sgn} (x)={\begin{cases}1\ ,&x>0\ ,\\0\ ,&x=0\ ,\\-1\ ,&x<0\ ,\end{cases}}}
x
∈
R
∖
{
0
}
{\displaystyle x\in \mathbb {R} \setminus \{0\}}
limes
lim
x
→
0
sgn
(
x
)
{\displaystyle \lim _{x\to 0}\,\operatorname {sgn} (x)}
Functio Dirichlet
f
:
R
→
R
,
f
(
x
)
:=
{
1
,
x
∈
Q
,
0
,
x
∈
R
∖
Q
,
{\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} \ ,\ f(x):=\left\{{\begin{array}{ll}1\ ,&x\in \mathbb {Q} \ ,\\0\ ,&x\in \mathbb {R} \setminus \mathbb {Q} \ ,\end{array}}\right.}
ubique discontinua est. Illustratio functionum discontinuarum:
Functio discontinua
Functio quidem discontinua, sed tamen continua de laevo latere
Functio quidem discontinua, sed tamen continua de dextro latere
Est defintio usitata:
Regula copiarum apertarum : Sint
X
{\displaystyle X}
Y
{\displaystyle Y}
f
:
X
→
Y
{\displaystyle f:X\rightarrow Y}
x
0
∈
X
{\displaystyle x_{0}\in X}
f
{\displaystyle f}
x
0
{\displaystyle x_{0}}
V
{\displaystyle V}
f
(
x
0
)
{\displaystyle f(x_{0})}
U
{\displaystyle U}
x
0
{\displaystyle x_{0}}
f
(
U
)
⊂
V
{\displaystyle f(U)\subset V}
[3] Si functiones
f
{\displaystyle f}
g
{\displaystyle g}
dominio communi
D
{\displaystyle D}
f
+
g
{\displaystyle f+g}
f
−
g
{\displaystyle f-g}
f
⋅
g
{\displaystyle f\cdot g}
D
{\displaystyle D}
x
0
∈
D
{\displaystyle x_{0}\in D}
g
(
x
0
)
=
0
{\displaystyle g(x_{0})=0}
f
g
{\displaystyle {\frac {f}{g}}}
Compositio
f
∘
g
{\displaystyle f\circ g}
Continuitas functionis inversae :Si
I
{\displaystyle I}
R
{\displaystyle \mathbb {R} }
f
:
I
→
R
{\displaystyle f\colon I\rightarrow \mathbb {R} }
I
{\displaystyle I}
f
{\displaystyle f}
J
{\displaystyle J}
f
:
I
→
J
{\displaystyle f\colon I\to J}
biiectiva , et functio inversa
f
−
1
:
J
→
I
{\displaystyle f^{-1}\colon J\to I}
Theorema valorum omnium acceptorum :Si
f
:
[
a
,
b
]
→
R
{\displaystyle f:[a,b]\to \mathbb {R} }
a
<
b
{\displaystyle a<b}
f
(
a
)
<
f
(
b
)
{\displaystyle f(a)<f(b)}
d
∈
[
f
(
a
)
,
f
(
b
)
]
{\displaystyle d\in [f(a),f(b)]}
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
f
(
x
)
=
d
{\displaystyle f(x)=d}
Item in casu
f
(
a
)
>
f
(
b
)
{\displaystyle f(a)>f(b)}
d
∈
[
f
(
b
)
,
f
(
a
)
]
{\displaystyle d\in [f(b),f(a)]}
Theorema extremitatum acceptarum :Si
f
:
[
a
,
b
]
→
R
{\displaystyle f\colon [a,b]\to \mathbb {R} }
t
,
h
∈
[
a
,
b
]
{\displaystyle t,h\in [a,b]}
f
(
t
)
≤
f
(
x
)
≤
f
(
h
)
{\displaystyle f(t)\leq f(x)\leq f(h)}
x
∈
[
a
,
b
]
{\displaystyle x\in [a,b]}
Nexus interni
↑ Harro Heuser, Lehrbuch der Analysis. Teil 1. 8. Auflage, B. G. Teubner, Stuttgart 1990. ISBN 3-519-12231-6 . Definition 34.6
↑ Harro Heuser, Lehrbuch der Analysis. Teil 1. 8. Auflage, B. G. Teubner, Stuttgart 1990. ISBN 3-519-12231-6 . S. 212
↑ Harro Heuser, Lehrbuch der Analysis. Teil 2. 14. Auflage, Teubner 2012. ISBN 3-835-10208-7 , pagina 230
![{\displaystyle {\begin{aligned}&\scriptstyle f\colon [-1,1.5]\to [-1,1.5]\\&\textstyle x\mapsto {\frac {(4x^{3}-6x^{2}+1){\sqrt {x+1}}}{3-x}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/367545798d4c1d7748410deb9d03ea9faf400d6d)
Functio discontinua
Functio quidem discontinua, sed tamen continua de laevo latere
Functio quidem discontinua, sed tamen continua de dextro latere
![{\displaystyle f:[a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b592d102ccd1ba134d401c5b3ea177baaba3ffac)
![{\displaystyle d\in [f(a),f(b)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e87457fce46840e85df4e4e35f069bf297f46c26)
![{\displaystyle x\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/026357b404ee584c475579fb2302a4e9881b8cce)
![{\displaystyle d\in [f(b),f(a)]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/86aa3db9f151ccbfdbdb64598664241ed45f8cc6)
![{\displaystyle f\colon [a,b]\to \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5ab61178bf5349838758ffe3d96135406ed0245)
![{\displaystyle t,h\in [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b1cf8393dd582a8a45cd3877d9ea1c04bf99665f)
