let X, Y be non empty TopSpace; :: thesis: for f being Function of X,Y

for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds

for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let f be Function of X,Y; :: thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds

for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X = X1 union X2 implies for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )

assume A1: X = X1 union X2 ; :: thesis: for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x be Point of X; :: thesis: for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x2 be Point of X2; :: thesis: ( x = x1 & x = x2 implies ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )

assume that

A2: x = x1 and

A3: x = x2 ; :: thesis: ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

thus ( f is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A3, Th58; :: thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x )

thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x ) :: thesis: verum

for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds

for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let f be Function of X,Y; :: thesis: for X1, X2 being non empty SubSpace of X st X = X1 union X2 holds

for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let X1, X2 be non empty SubSpace of X; :: thesis: ( X = X1 union X2 implies for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )

assume A1: X = X1 union X2 ; :: thesis: for x being Point of X

for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x be Point of X; :: thesis: for x1 being Point of X1

for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x1 be Point of X1; :: thesis: for x2 being Point of X2 st x = x1 & x = x2 holds

( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

let x2 be Point of X2; :: thesis: ( x = x1 & x = x2 implies ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) )

assume that

A2: x = x1 and

A3: x = x2 ; :: thesis: ( f is_continuous_at x iff ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) )

thus ( f is_continuous_at x implies ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 ) ) by A2, A3, Th58; :: thesis: ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x )

thus ( f | X1 is_continuous_at x1 & f | X2 is_continuous_at x2 implies f is_continuous_at x ) :: thesis: verum

proof

assume that

A4: f | X1 is_continuous_at x1 and

A5: f | X2 is_continuous_at x2 ; :: thesis: f is_continuous_at x

for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G

end;A4: f | X1 is_continuous_at x1 and

A5: f | X2 is_continuous_at x2 ; :: thesis: f is_continuous_at x

for G being a_neighborhood of f . x ex H being a_neighborhood of x st f .: H c= G

proof

hence
f is_continuous_at x
; :: thesis: verum
let G be a_neighborhood of f . x; :: thesis: ex H being a_neighborhood of x st f .: H c= G

f . x = (f | X1) . x1 by A2, FUNCT_1:49;

then consider H1 being a_neighborhood of x1 such that

A6: (f | X1) .: H1 c= G by A4;

the carrier of X1 c= the carrier of X by BORSUK_1:1;

then reconsider S1 = H1 as Subset of X by XBOOLE_1:1;

f . x = (f | X2) . x2 by A3, FUNCT_1:49;

then consider H2 being a_neighborhood of x2 such that

A7: (f | X2) .: H2 c= G by A5;

the carrier of X2 c= the carrier of X by BORSUK_1:1;

then reconsider S2 = H2 as Subset of X by XBOOLE_1:1;

f .: S2 c= G by A7, FUNCT_2:97;

then A8: S2 c= f " G by FUNCT_2:95;

consider H being a_neighborhood of x such that

A9: H c= H1 \/ H2 by A1, A2, A3, Th16;

take H ; :: thesis: f .: H c= G

f .: S1 c= G by A6, FUNCT_2:97;

then S1 c= f " G by FUNCT_2:95;

then S1 \/ S2 c= f " G by A8, XBOOLE_1:8;

then H c= f " G by A9, XBOOLE_1:1;

hence f .: H c= G by FUNCT_2:95; :: thesis: verum

end;f . x = (f | X1) . x1 by A2, FUNCT_1:49;

then consider H1 being a_neighborhood of x1 such that

A6: (f | X1) .: H1 c= G by A4;

the carrier of X1 c= the carrier of X by BORSUK_1:1;

then reconsider S1 = H1 as Subset of X by XBOOLE_1:1;

f . x = (f | X2) . x2 by A3, FUNCT_1:49;

then consider H2 being a_neighborhood of x2 such that

A7: (f | X2) .: H2 c= G by A5;

the carrier of X2 c= the carrier of X by BORSUK_1:1;

then reconsider S2 = H2 as Subset of X by XBOOLE_1:1;

f .: S2 c= G by A7, FUNCT_2:97;

then A8: S2 c= f " G by FUNCT_2:95;

consider H being a_neighborhood of x such that

A9: H c= H1 \/ H2 by A1, A2, A3, Th16;

take H ; :: thesis: f .: H c= G

f .: S1 c= G by A6, FUNCT_2:97;

then S1 c= f " G by FUNCT_2:95;

then S1 \/ S2 c= f " G by A8, XBOOLE_1:8;

then H c= f " G by A9, XBOOLE_1:1;

hence f .: H c= G by FUNCT_2:95; :: thesis: verum