1. ## Riemann integrability

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

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

are Riemann integrable on [0,1].

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

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

are Riemann integrable on [0,1].

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

3. yeah, sorry, I forgot the other parenthesis

4. Originally Posted by thaopanda
yeah, sorry, I forgot the other parenthesis
Well $\displaystyle \sin(1/2)$ is just a number. Call it $\displaystyle k$. So you need to know for which n is $\displaystyle f_n(x)=kx^n$ integrable on $\displaystyle [0,1]$.

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

5. if it was $\displaystyle sin(\frac{1}{x})$, how would that change the problem?

6. Originally Posted by thaopanda
if it was $\displaystyle sin(\frac{1}{x})$, how would that change the problem?
$\displaystyle x^n\sin(1/x)$ is continuous for all $\displaystyle n\geq1$. If $\displaystyle n=0$, the function is still continuous on $\displaystyle (0,1]$ and is therefore integrable.