Understanding definition of Riemann Integral I have read this definition first time today. As far as I can understand it, it seems to me that difference between Riemann sums and a number L can be made small  by changing norm of partition. Does the changing norm of partition change values of $L$ and $\varepsilon$? $\delta$ is dependent on $\varepsilon$, HOW? What does it mean that the set of all Riemann integrable functions on $[a,b]$ is $\mathcal R[a,b]$?
Thanks for help 

 A: Just follow carefully on the order of quantifications. The function is Reimann integrable if there exists a number $L$ (and it remains fixed!) such that for any $\varepsilon >0$ (which you are free to choose as you like, but once you made that choice it remains fixed) there exists a $\delta_\varepsilon >0$ (which may depend on $\varepsilon $ in any way whatsoever (and typically it does not matter at all how it depends on it)) such that for any partition with mesh less than $\delta_\varepsilon $, the corresponding Riemann sum is within distance $\varepsilon $ from $L$. Finally, $\mathcal R[a,b]$ is just the name given to the set of all Riemann integrable functions on $[a,b]$. 
A: If your function is continuous on the interval $[a,b]$ it is also uniformly continuous:
$$\forall, \epsilon > 0 \; \exists \delta>0: |x-y|< \delta \; x,y \in [a,b]\Rightarrow |f(x)-f(y)|<\epsilon $$
take the norm of your partitions($P_1,P_2$) to be less than $\delta$. Let $P_3$ refine both partitions $P_3 = P_1 \cup P_2$
$$S(f,P_1) - S(f,P_2) = \sum_{j=1}^N (f(x_{t_j})- f(y_{t_j}))(t_{j+1}-t_j) $$
Therefore 
\begin{align*}
|S(f,P_1) - S(f,P_2)| &= \sum_{j=1}^N |(f(x_{t_j})- f(y_{t_j}))|(t_{j+1}-t_j) \\
&\leq \sum_{j=1}^N \epsilon  (t_{j+1}-t_j) = \epsilon (b-a)
\end{align*}
So as you reduce the norm of your partition, the distance to the limit of your sum gets smaller (at least an upper bound to this distance).
