# Riemann Integrable

• Jan 19th 2010, 08:13 PM
Chandru1
Riemann Integrable
Suppose $f$ is a riemann integrable function on $[a,b].$ If $g$ is a bounded real valued function defined on $[a,b]$ and $g$ differs from $f$ at infinitely many points in $[a,b]$, should $g$ be riemann integrable on $[a,b]$. Prove or disprove?
• Jan 19th 2010, 08:21 PM
Drexel28
Quote:

Originally Posted by Chandru1
Suppose $f$ is a riemann integrable function on $[a,b].$ If $g$ is a bounded real valued function defined on $[a,b]$ and $g$ differs from $f$ at infinitely many points in $[a,b]$, should $g$ be riemann integrable on $[a,b]$. Prove or disprove?

This makes no sense. Let $f(x)=3$ and $g(x)=\begin{cases} 1 & \mbox{if} \quad x\in\mathbb{Q}\\ 0 & \mbox{if} \quad x\notin\mathbb{Q}\end{cases}$
• Jan 20th 2010, 02:03 AM
Shanks
It shoud be "g differs from f at finitely many points in [a,b]", I think.
• Jan 20th 2010, 08:14 AM
Drexel28
Quote:

Originally Posted by Chandru1
Suppose $f$ is a riemann integrable function on $[a,b].$ If $g$ is a bounded real valued function defined on $[a,b]$ and $g$ differs from $f$ at infinitely many points in $[a,b]$, should $g$ be riemann integrable on $[a,b]$. Prove or disprove?

Quote:

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?