let X, Y be non empty TopSpace; for X1, X2 being non empty SubSpace of X
for g being Function of (X1 union X2),Y
for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let X1, X2 be non empty SubSpace of X; for g being Function of (X1 union X2),Y
for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let g be Function of (X1 union X2),Y; for x1 being Point of X1
for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x1 be Point of X1; for x2 being Point of X2
for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x2 be Point of X2; for x being Point of (X1 union X2) st x = x1 & x = x2 holds
( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
let x be Point of (X1 union X2); ( x = x1 & x = x2 implies ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) ) )
assume that
A1:
x = x1
and
A2:
x = x2
; ( g is_continuous_at x iff ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
A3:
X2 is SubSpace of X1 union X2
by TSEP_1:22;
A4:
X1 is SubSpace of X1 union X2
by TSEP_1:22;
hence
( g is_continuous_at x implies ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 ) )
by A1, A2, A3, Th74; ( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x )
thus
( g | X1 is_continuous_at x1 & g | X2 is_continuous_at x2 implies g is_continuous_at x )
verumproof
assume that A5:
g | X1 is_continuous_at x1
and A6:
g | X2 is_continuous_at x2
;
g is_continuous_at x
for
G being
a_neighborhood of
g . x ex
H being
a_neighborhood of
x st
g .: H c= G
proof
let G be
a_neighborhood of
g . x;
ex H being a_neighborhood of x st g .: H c= G
g . x = (g | X1) . x1
by A1, A4, Th65;
then consider H1 being
a_neighborhood of
x1 such that A7:
(g | X1) .: H1 c= G
by A5;
g . x = (g | X2) . x2
by A2, A3, Th65;
then consider H2 being
a_neighborhood of
x2 such that A8:
(g | X2) .: H2 c= G
by A6;
the
carrier of
X2 c= the
carrier of
(X1 union X2)
by A3, TSEP_1:4;
then reconsider S2 =
H2 as
Subset of
(X1 union X2) by XBOOLE_1:1;
g .: S2 c= G
by A3, A8, Th68;
then A9:
S2 c= g " G
by FUNCT_2:95;
the
carrier of
X1 c= the
carrier of
(X1 union X2)
by A4, TSEP_1:4;
then reconsider S1 =
H1 as
Subset of
(X1 union X2) by XBOOLE_1:1;
consider H being
a_neighborhood of
x such that A10:
H c= H1 \/ H2
by A1, A2, Th16;
take
H
;
g .: H c= G
g .: S1 c= G
by A4, A7, Th68;
then
S1 c= g " G
by FUNCT_2:95;
then
S1 \/ S2 c= g " G
by A9, XBOOLE_1:8;
then
H c= g " G
by A10, XBOOLE_1:1;
hence
g .: H c= G
by FUNCT_2:95;
verum
end;
hence
g is_continuous_at x
;
verum
end;