Question

Property of continuous function

Answer 1

If \[f \in C[0,1], f: [0,1] \rightarrow R\] does it hold \[(f+g)(x)=f(x)+g(x), x \in [0,1]\]?

Answer 2

what is the definition of \((f+g)(x)?\)

Answer 3

\[g \in C[0,1] too\]

Answer 4

ok i think maybe the question is asking this
correct me if i am wrong

"if \(f\) and \(g\) are continuous on \([0,1]\) is \(f+g\) also continuous on \([0,1]\) ?"

Answer 5

i.e. "is the sum of two continuous functions also continuous?"

Answer 6

Not really. I actually have to prove that mapping \[A(f)(x)=(x^4-x^2)f(x), x \in [0,1], f \in C[0,1]\] is linear. I thought that I had to prove that it holds\[A(\alpha f+\beta g)(x)=\alpha A(f)(x)+\beta A(g)(x), \alpha,\beta\in R\]

Answer 7

ooh i see
ok that should be ok right? you can assume that it is continuous because the sum and product of continuous functions is a continuous function
what you have to do is compute the left side \[A(\alpha f+\beta g)(x)\] and show that it is equal to the right side \[\alpha A(f)(x)+\beta A(g)(x)\]

Answer 8

I have problem there. I have \[A(\alpha f+\beta g)(x)=(x^4-x^2)(\alpha f+\beta g)(x)\]now I don't know if \[(\alpha f+\beta g)(x)=\alpha f(x)+\beta g(x)\] If it is, why is it?

Answer 9

yes by the definition of adding functions

Answer 10

Thank you :)

Answer 11

what does \(f+g\) mean? it means take \(f(x)+g(x)\) right?

Answer 12

Yes, I guess

Answer 13

Is it correct to write (f+g)(x)?