reserve n,m,k for Nat;
reserve x,X,X1 for set;
reserve r,p for Real;
reserve s,g,x0,x1,x2 for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of REAL,the carrier of S;
reserve s1,s2 for Real_Sequence;
reserve Y for Subset of REAL;

theorem
f is total & (for x1,x2 holds f/.(x1+x2) = f/.x1 + f/.x2)
  & (ex x0 st f is_continuous_in x0)
implies f|REAL is continuous
proof
   assume that
A1: f is total and
A2: for x1,x2 holds f/.(x1+x2) = f/.x1 + f/.x2;
A3:dom f = REAL by A1,PARTFUN1:def 2;
   given x0 such that
A4: f is_continuous_in x0;
   reconsider z0=0 as Real;
   f/.z0 + f/.z0 = f/.(z0+z0) by A2; then
   f/.z0 + (f/.z0 - f/.z0) = f/.z0 - f/.z0 by RLVECT_1:28; then
   f/.z0 + 0.S = f/.z0 - f/.z0 by RLVECT_1:15; then
A5:f/.z0 + 0.S = 0.S by RLVECT_1:15;
A6:now let x1;
    0.S = f/.(x1+-x1) by A5,RLVECT_1:4; then
    0.S = f/.x1+f/.(-x1) by A2;
    hence -(f/.x1)=f/.(-x1) by RLVECT_1:def 10;
   end;
A7:now let x1,x2;
    f/.(x1-x2) = f/.(x1+-x2); then
    f/.(x1-x2) = f/.x1 + f/.(-x2) by A2;
    hence f/.(x1-x2) = f/.x1 - f/.x2 by A6;
   end;
   now let x1,r;
    assume that
     x1 in REAL and
A8:  r>0;
    set y=x1-x0;
    consider s such that
A9:  0<s and
A10: for x1 st x1 in dom f & |.x1-x0.|<s holds ||. f/.x1 - f/.x0 .||<r
        by A4,A8,Th8;
    take s;
    thus s>0 by A9;
    let x2 such that
     x2 in REAL and
A11: |.x2-x1.|<s;
A12: x2-y in REAL & |.x2-y-x0.| = |.x2-x1.| by XREAL_0:def 1;
    y+x0=x1; then
    ||.f/.x2-f/.x1.|| = ||.f/.x2-(f/.y+f/.x0).|| by A2
      .= ||.f/.x2-f/.y-f/.x0 .|| by RLVECT_1:27
      .= ||.f/.(x2-y)-f/.x0 .|| by A7;
    hence ||.f/.x2-f/.x1.||<r by A3,A10,A11,A12;
   end;
   hence thesis by A3,Th17;
end;
