1. ## Riemann Integrable

Suppose $\displaystyle f$ is a riemann integrable function on $\displaystyle [a,b].$ If $\displaystyle g$ is a bounded real valued function defined on $\displaystyle [a,b]$ and $\displaystyle g$ differs from $\displaystyle f$ at infinitely many points in $\displaystyle [a,b]$, should $\displaystyle g$ be riemann integrable on $\displaystyle [a,b]$. Prove or disprove?

2. Originally Posted by Chandru1
Suppose $\displaystyle f$ is a riemann integrable function on $\displaystyle [a,b].$ If $\displaystyle g$ is a bounded real valued function defined on $\displaystyle [a,b]$ and $\displaystyle g$ differs from $\displaystyle f$ at infinitely many points in $\displaystyle [a,b]$, should $\displaystyle g$ be riemann integrable on $\displaystyle [a,b]$. Prove or disprove?
This makes no sense. Let $\displaystyle f(x)=3$ and $\displaystyle g(x)=\begin{cases} 1 & \mbox{if} \quad x\in\mathbb{Q}\\ 0 & \mbox{if} \quad x\notin\mathbb{Q}\end{cases}$

3. It shoud be "g differs from f at finitely many points in [a,b]", I think.

4. Originally Posted by Chandru1
Suppose $\displaystyle f$ is a riemann integrable function on $\displaystyle [a,b].$ If $\displaystyle g$ is a bounded real valued function defined on $\displaystyle [a,b]$ and $\displaystyle g$ differs from $\displaystyle f$ at infinitely many points in $\displaystyle [a,b]$, should $\displaystyle g$ be riemann integrable on $\displaystyle [a,b]$. Prove or disprove?
Originally Posted by Shanks
It shoud be "g differs from f at finitely many points in [a,b]", I think.
Is that so? Or is it countably many. Or what?