つづき

History
In his 1821 book Cours d'analyse, Augustin-Louis Cauchy discussed variable quantities, infinitesimals and limits, and defined continuity of
y=f(x) by saying that an infinitesimal change in x necessarily produces an infinitesimal change in y, while Grabiner claims that he used a rigorous epsilon-delta definition in proofs.[4]
In 1861, Karl Weierstrass first introduced the epsilon-delta definition of limit in the form it is usually written today.[5] He also introduced the notations
lim and
lim x→x0.[6]
The modern notation of placing the arrow below the limit symbol is due to G. H. Hardy, which is introduced in his book A Course of Pure Mathematics in 1908.[7]

Functions on metric spaces
Suppose M and N are subsets of metric spaces A and B, respectively, and f : M → N is defined between M and N, with x ∈ M, p a limit point of M and L ∈ N. It is said that the limit of f as x approaches p is L and write
lim x→p f(x)=L
if the following property holds:
For every ε > 0, there exists a δ > 0 such that for all points x ∈ M, 0 < dA(x, p) < δ implies dB(f(x), L) < ε.[22]
(∀ε>0)(∃δ>0)(∀x∈M)(0<d A(x,p)<δ⟹dB(f(x),L)<ε).
Again, note that p need not be in the domain of f, nor does L need to be in the range of f, and even if f(p) is defined it need not be equal to L.
Euclidean metric
略
(引用終り)
以上