Thursday, 9 March 2017

calculus - Riemann integrable over given interval?

I was looking for a brief explanation as to which of the following functions are Riemann integrable or not.




$f(x):=(1-x^2)^{-1} $if $ x≠ 1 $ and $ x ≠1,$



$f(x):= 0, $ if $ x=1 $ or $ x=-1.$



(Over interval: $[-1,1]$)



I can see that the function is continuous & hence Riemann integrable over $(-1,1)$ but don't know how I can explain that it is over $[-1,1]$. I am thinking I maybe can just say that it is bounded?



Also




$f(x)=\frac{1}{x^2-3x-4},$ over interval $[-3,0]$,
I see in the interval when $x=-1$, $f(x)$ is undefined. Is it acceptable to say that this function is hence not bounded over this interval and hence not Riemann integrable?

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