Calculating Functional Derivatives: -1≤xₒ≤1 vs -1<xₒ<1

[PBTR3]

TL;DR

A functional derivative using the delta function with intergral limits of +-1.

$\frac {\delta I[f]} {\delta f(x_o)} = \int_a ^b \delta(x-x_o) \, dx$ with a=-1 and b=+1

$-1 \leq x_o \leq +1$ vs $-1 \lt x_o \lt +1$, 0 otherwise. Which is correct and does it matter when doing integration by parts?

 

[PBTR3]

I could not get the Latex Preview to work until now. This is more like what I intended.