Please help me go about this problem
Thank you
Let f(x) = {1 0<=x<1 and 2 x=1} and let P = {0,x1,x2,...,x(n-1),1} be a partition of [0,1]. Compute L(P,,f) and U(P,f). Your answers may contain terms from the partition
You may know this but that are almost as many definitions & notations for Riemann integrals as there are people who have written about the integral. Thus you must post the definitions & notations your text material is using.
What does it mean for the integral to exist? What is the difference in $L(f.P)~\&~L(f)~?$
@ aris27, I asked that you post the definitions and theorems that you are using. Either you don’t know or too lazy to do so. Since I have done text material on these, here is my take.
If $\mathcal{P}=\{x_0,x_1,\cdots,x_n\}$ is a finite increasing sequence of points in the interval $[a.b]$ such that $x_0=a~\&~x_n=b$ then $\mathcal{P}$ is a partition of $[a,b]$ Now if $Q$ is also a partition of $[a.b]$ such that each $[y_k,y_{k+1}]$ in $\mathcal{Q}$ is a subset of some $[x_j.x_{j-1}]$ in $\mathcal{P}$ then $\mathcal{Q}$ is a refinement of $\mathcal{P}$.
Now suppose that $f$ is a bounded function from $[a,b]$ and $\mathcal{P}$ is a partition of $[a,b]$, then $(\forall k)[0 \le k\le n-1]$ we define $m_k=\min\{f(x):k\le x\le x_{k+1}\}$ likewise $M_k=\max\{f(x):k\le x\le x_{k+1}\}$.
So $L(f,\mathcal{P}=\sum\limits_{j = 0}^{n - 1} {{m_j}\left( {{x_{j + 1}} - {x_j}} \right)}$. Likewise, $U(f,\mathcal{P}=\sum\limits_{j = 0}^{n - 1} {{M_j}\left( {{x_{j + 1}} - {x_j}} \right)}$.
Now suppose that $\displaystyle\lim {\sup _P}L(f,P) = \int_a^b f = \lim {\inf _P}U(f,P)$ then the function $f$ is Riemann intergrable on $[a,b]$