Functions. Limits and Continuity. Arcs and Curves 149
Hence
d'(f(x),l(x')):::; d'(f(x),l(x;)) + d'(f(x;),l(x'))
< l/2€ + l/2€ = €
so l is uniformly continuous on E.
Definition 3.13 Let (X, d) and (Y, d') be t~o metric spaces. A function
l: X -+ Y is said to be sequentially continuous at x E X if for any
{ Xn} C X_ such ~hat Xn -+ X We have l ( Xn) -+ l ( X ).
Theorem 3.12 Let j: X -+ Y. Then l is continuous at x E X iff l is
sequentially continuous at x.
Proof Suppose that l is continuous at x and that Xn -+ x. Then for every
E > 0 there is a 8 > 0 such that if x' E N6(x) we have d'(f(x),f(x')) < E.
Since Xn -+ x, there exists N such that n 2: N implies that d( Xn, x) < 8,
or Xn E N6(x). Hence it follows that d'(f(x), l(xn)) < E, showing that
l(xn) -+ l(x).
Now suppose that l(xn)-+ l(x) for any sequence Xn-+ x, and assume
that l is not continuous at x. Then there exists a neighborhood N<(f(x))
such that an arbitrary N6(x) contains points x' E X such that l(x') rj.
N<(f(x)), i.e., such that d'(f(x),l(x')) 2: E. Choosing On = l/n (n =
1,2, ... ), select in each N6n(x) a point Xn such that l(xn) rj. N<(f(x)).
Then Xn -+ x but {f(xn)} does not converge to l(x), contradicting the
hypothesis that l is sequentially continuous.
Definition3.14 Thefunctionl: X-+ Yfromametricspaceintoanother
(more generally, from a topological space into another) is called an open
function (or an interior mapping) if for any open set A C X the image
l(A) is open in Y. If l is an open mapping and one-to one, it is clear that
the inverse function 1-l exists and is continuous since u-^1 )-^1 = l maps
open sets into open sets.
The function l: X -+ Y is said to be open at a point x E X (also,
locally interior at x E X) if for every open set A C X containing x, l ( x) is
an interior point of l(A). Clearly, a function is open on X iff it is locally
interior everywhere ~n X.
Definition 3.15 A one-to-one function l between topological spaces is
called a homeomorphism if l and 1-^1 are both continuous. Functions
having the last property, i.e., such that both l and l-^1 are continuous,
are sometimes called bicontinuous.
The spaces X and Y are said to be homeomorphic (or topologically equiv-
alent) if a homeomorphic correspondence can be established between them.
