I was stuck when proving a theorem in Introduction to real analysis (4th edition).
I don't know why the author assumed $c=x_{i}=x_{i-1}$. Is it because $x_{i}$ and $x_{i+1}$ are close enough or coincide. But if so, the mesh $\| \dot{\mathcal{P}} \|$ should be zero then there is no need to take that into account. Could anyone explain why this proof did so?
By the way, the definition of partition is below:
The author doesn't assume that $c = x_i =x_{i-1}$. He says that $S(f, \dot{\mathcal P})$ and $S(g, \dot{\mathcal P})$ are equal except if $c$ is one of the endpoint of the tagged partition $\dot{\mathcal P}$. And in the later case he bounds the difference$S(f, \dot{\mathcal P})-S(g, \dot{\mathcal P})$.
Also, I think that the author meant $c=x_i$ or $c=x_{i-1}$. $c = x_i = x_{i-1}$ doesn't make sense as the endpoints of a partition are supposed to be distinct.