for part A i assume that f is in norm space C[0,1],||.|| , then choose a sequence fn in C[o,1] s.t fn->f then for 0<fn<1 so 0<f<1 i.e. A is closed i am not sure my answer here

for part B i assume the anti-derivatice of f(t) to be K(t)+c therefore, by F(f)=2K(1/2)+1/2-K(1)-K(0) then how should i prove it is cts ???