# A question related to properties of Upper and Lower Riemann Sums

This particular question was asked in my Analysis quiz and I was unable to solve it . I have studied analysis from Tom M Apostol .

Let $$f$$ be a continuously differentiable real valued function on $$[a,b]$$ such that $$|f'(x)|\leq K$$ for all $$x \in [a,b]$$. For a partition $$P=\{a=a_{0} let $$U(f,P)$$ and $$L(f,P)$$ denote the Upper and Lower Riemann Sums of $$f$$ with respect to $$P$$. Then which one of the following is correct?

A. $$|L(f,P)|\leq K(b-a)\leq |U(f,P)|$$.

B. $$U(f,P)-L(f,P) \leq K(b-a)$$.

C. $$U(f, P)- L(f,P) \leq K\|P\|$$, where $$\|P\| =\max_{0\le i\le n-1} (a_{i+1} - a_{i} )$$.

D. $$U(f,P)-L(f,P) \leq K\|P\|(b-a)$$.

Finding counterexamples to this would be tedious. So, I think its better to look for the correct option. But I am really confused.

Can anyone please tell how should I approach the problem.

$$U(f,P)-L(f,P)$$ $$=\sum_i \sup |f(x(i))-f(y(i))||x_{i+1}-x_i|$$ (I use $$x(i)$$ to show dependence of $$x$$ on $$i$$, because $$x(i),y(i)\in [x_i,x_{i+1}]$$
This equals $$\sum_i \frac{|f(x)-f(y)|}{|x-y|}|x_{i+1}-x_i||x-y|$$ (I assume the difference is maximised at $$x,y$$) which equals $$\sum_i |f'(\xi_i)||x_{i+1}-x_i||x-y|$$ for some $$\xi_i \in (x_{i},x_{i+1})$$ and hence this is $$\leq K||P||(b-a)$$