Integral defined as a limit using regular partitions Definition. Given a function $f$ defined on $[a,b]$, let $$\xi_k \in [x_{k-1},x_k],\quad k=1,\ldots,n$$ where $$ x_k=a+k\frac{b-a}n, \quad k=0,\ldots,n \; .$$
One says that $f$ is integrable on $[a,b]$ if the limit $$\lim_{n\to\infty}\frac{b-a}n\sum_{k=1}^n f(\xi_k)$$ 
exists and is independent of the $\xi_k$.
I seek a proof of the:
Theorem. If $a<c<b$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b].$

HINT:
Take Two cases:
1.When $c$ is a tag of a sub-interval $[x_{k},x_{k+1}]$ of $\dot{P}$,where $\dot{P}$ is your tagged partition ${(I_{i},t_{i})}_{i=1}^{n}$,such that $I_{i}=[x_{i},x_{i+1}]$
2.When $c$ is an end-point of a sub-interval of $\dot{P}$.
