Riemann integrability

**thaopanda** · December 6th 2009, 05:07 PM

Decide which of the functions $f_n$ : [0,1] $\rightarrow$ R with n = 0,1,2, defined by setting:

$f_n(x)$ :=
$(x^n)sin(\frac{1}{2}$ if $0 < x \leq 1$
0 if x = 0

are Riemann integrable on [0,1].

more help please

**redsoxfan325** · December 6th 2009, 08:54 PM

Originally Posted by thaopanda

Decide which of the functions $f_n$ : [0,1] $\rightarrow$ R with n = 0,1,2, defined by setting:

$f_n(x)$ :=
$(x^n)sin(\frac{1}{2}$ if $0 < x \leq 1$
0 if x = 0

are Riemann integrable on [0,1].

more help please

Is that $f_n(x)=x^n\sin\left(\frac{1}{2}\right)<br />$ ?

**thaopanda** · December 7th 2009, 04:46 AM

yeah, sorry, I forgot the other parenthesis

**redsoxfan325** · December 7th 2009, 06:40 AM

Originally Posted by thaopanda

yeah, sorry, I forgot the other parenthesis

Well $\sin(1/2)$ is just a number. Call it $k$ . So you need to know for which n is $f_n(x)=kx^n$ integrable on $[0,1]$ .

For $n\geq1$ , it's continuous and therefore integrable. For $n=0$ , you have one removable discontinuity (at $x=0$ ), so it's also integrable.