Convergence theorem2
Logic
Propositional logic
Predicate-Logic
Set
Set
Set operation
Binary relation and Map
Equivalence relation
Ordered set
Wellordered set
Axiom of Choice
Cardinality
Topology
Topology
Topological space
Continuous map
Compact
Separation axioms
Metric-space
Convergence
Linear algebra
Linear space
Product space
Basis
Orthogonal system
Linear operator
Determinant
Elementary matrix
Rank
Eigenspace
Nomal matrix
Group
Group
Subgroup
Cyclic group
Symmetric group
Quotient group
Homomorphism
Cartesian product
Group action
Real analysis
Convergence of real sequence
Convergence theorem1
Convergence of real series
Convergence of real function
Convergence theorem2
Measure theory
Finitely additive measure
Outer measure
Measure space
Back
en
ja
Continuous real function
f
(
x
)
is continuous at
a
:
x
→
a
lim
f
(
x
)
=
f
(
a
)
Left, right continuous
f
(
x
)
is left continuous at
a
:
(
a
,
a
+
ε
0
)
⊂
I
x
→
a
+
0
lim
f
(
x
)
=
f
(
a
)
f
(
x
)
is right continuous
a
:
(
a
−
ε
0
,
a
)
⊂
I
x
→
a
−
0
lim
f
(
x
)
=
f
(
a
)
Theorem
f
(
x
)
is continous at
a
⇔
x
→
a
+
0
lim
f
(
x
)
=
f
(
a
)
∧
x
→
a
−
0
lim
f
(
x
)
=
f
(
a
)
Continuous real function
Left, right continuous
Theorem
