# Thread: Algebra with Integral functions

1. ## Algebra with Integral functions

I am trying to prove that if $\displaystyle a<c<b$, then $\displaystyle \int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f$. This fact is very easy for me to visualize but I'm not sure how to get started.
My professor talked about restricting the function and only evaluating the integral on a smaller interval. I feel like this might be key to proving it but I'm not sure how to tie it together. Any help would be appreciated.

2. Originally Posted by zebra2147
I am trying to prove that if $\displaystyle a<c<b$, then $\displaystyle \int _{a}^{b}f=\int _{a}^{c}f=\int _{c}^{b}f$.
The is a typo in your post. It must be $\displaystyle \int _{a}^{b}f=\int _{a}^{c}f+\int _{c}^{b}f$
Consider a partition $\displaystyle \mathcal{P}$ of $\displaystyle [a,c]$ and a partition $\displaystyle \mathcal{Q}$ of $\displaystyle [c,b]$.
Then $\displaystyle \mathcal{P}\cup\mathcal{Q}$ is a partition of $\displaystyle [a,b].$

On the other hand, if a partition $\displaystyle \mathcal{R}$ of $\displaystyle [a,b]$ does not contain $\displaystyle c$ as a division point then simply add as a division point . Now you have two partitions, one of $\displaystyle [a,c]$ and one of $\displaystyle [c,b]$.

That is an outline if an if and only if proof.