riemann integrable functions map to another one
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.Doe
62
add a comment |
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.Doe
62
add a comment |
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
asked Dec 11 '18 at 21:53
J.Doe
62
I have the following question to prove:
If $f_{1}, f_{2},...,f_{n}$ are Riemann integrable functions, and $phi$ a function from $R^{n}$ to $R$, where $R$ denotes the set of real numbers, such that $phi$ is monotone with respect to each of its coordinates, prove that $phi(f_1,...,f_n)$ is also Riemann integrable.
I had the following: approach $phi(f_1,...,f_n)$ with $phi(psi_1,...,psi_n)$ where $psi_{i}$ is a step function that controls $f_i$ however I have some trouble controlling the difference without any hypothesis regarding the linearity.
EDIT: I'm currently trying with $n = 1$ , aka $phi$ is a monotone function. One thing that I know is that it has countable discontinuity points.
Could i get some hints? THanks in advance
analysis
analysis
asked Dec 11 '18 at 21:53
J.Doe
62
asked Dec 11 '18 at 21:53
J.Doe
62
edited Dec 12 '18 at 13:31
asked Dec 11 '18 at 21:53
J.Doe
62
asked Dec 11 '18 at 21:53
J.Doe
62
asked Dec 11 '18 at 21:53
J.Doe
62
62
add a comment |
add a comment |
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