Other characterizations Limit of a function

1 other characterizations

1.1 in terms of sequences
1.2 in non-standard calculus
1.3 in terms of nearness

other characterizations
in terms of sequences

for functions on real line, 1 way define limit of function in terms of limit of sequences. (this definition attributed eduard heine.) in setting:

lim

x
→
a


f
(
x
)
=
l


{\displaystyle \lim _{x\to a}f(x)=l}

if , if sequences




x

n




{\displaystyle x_{n}}

(with




x

n




{\displaystyle x_{n}}

not equal n) converging



a


{\displaystyle a}

sequence



f
(

x

n


)


{\displaystyle f(x_{n})}

converges



l


{\displaystyle l}

. shown sierpiński in 1916 proving equivalence of definition , definition above, requires , equivalent weak form of axiom of choice. note defining means sequence




x

n




{\displaystyle x_{n}}

converge



a


{\displaystyle a}

requires epsilon, delta method.

similarly case of weierstrass s definition, more general heine definition applies functions defined on subsets of real line. let f real-valued function domain dm(f). let limit of sequence of elements of dm(f). limit (in sense) of f l x approaches p if every sequence




x

n




{\displaystyle x_{n}}

 ∈ dm(f) \ {a} (so n,




x

n




{\displaystyle x_{n}}

not equal a) converges a, sequence



f
(

x

n


)


{\displaystyle f(x_{n})}

converges



l


{\displaystyle l}

. same definition of sequential limit in preceding section obtained regarding subset dm(f) of r metric space induced metric.

in non-standard calculus

in non-standard calculus limit of function defined by:

lim

x
→
a


f
(
x
)
=
l


{\displaystyle \lim _{x\to a}f(x)=l}

if , if



x
∈


r


∗




{\displaystyle x\in \mathbb {r} ^{*}}

,




f

∗


(
x
)
−
l


{\displaystyle f^{*}(x)-l}

infinitesimal whenever



x
−
a


{\displaystyle x-a}

infinitesimal. here





r


∗




{\displaystyle \mathbb {r} ^{*}}

hyperreal numbers ,




f

∗




{\displaystyle f^{*}}

natural extension of f non-standard real numbers. keisler proved such hyperreal definition of limit reduces quantifier complexity 2 quantifiers. on other hand, hrbacek writes definitions valid hyperreal numbers must implicitly grounded in ε-δ method, , claims that, pedagogical point of view, hope non-standard calculus done without ε-δ methods cannot realized in full. bŀaszczyk et al. detail usefulness of microcontinuity in developing transparent definition of uniform continuity, , characterize hrbacek s criticism dubious lament .

in terms of nearness

at 1908 international congress of mathematics f. riesz introduced alternate way defining limits , continuity in concept called nearness . point



x


{\displaystyle x}

defined near set



a
⊆

r



{\displaystyle a\subseteq \mathbb {r} }

if every



r
>
0


{\displaystyle r>0}

there point



a
∈
a


{\displaystyle a\in a}






|

x
−
a

|

<
r


{\displaystyle |x-a|<r}

. in setting the

lim

x
→
a


f
(
x
)
=
l


{\displaystyle \lim _{x\to a}f(x)=l}

if , if



a
⊆

r



{\displaystyle a\subseteq \mathbb {r} }

,



l


{\displaystyle l}

near



f
(
a
)


{\displaystyle f(a)}

whenever



a


{\displaystyle a}

near



a


{\displaystyle a}

.

here



f
(
a
)


{\displaystyle f(a)}

set



{
f
(
x
)

|

x
∈
a
}


{\displaystyle \{f(x)|x\in a\}}

. definition can extended metric , topological spaces.