Riemann Integrable

**Chandru1** · January 19th 2010, 08:13 PM

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?

**Drexel28** · January 19th 2010, 08:21 PM

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 $g(x)=\begin{cases} 1 & \mbox{if} \quad x\in\mathbb{Q}\\ 0 & \mbox{if} \quad x\notin\mathbb{Q}\end{cases}$

**Shanks** · January 20th 2010, 02:03 AM

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

**Drexel28** · January 20th 2010, 08:14 AM

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?

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?

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