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

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SUMMARY

The discussion focuses on the calculation of functional derivatives, specifically comparing the cases of -1 ≤ xₒ ≤ 1 and -1 < xₒ < 1. The integral representation of the functional derivative is given as δI[f]/δf(xₒ) = ∫_a^b δ(x - xₒ) dx with limits a = -1 and b = +1. The participants debate the implications of the boundary conditions on integration by parts, concluding that the choice of boundary conditions is significant for the correctness of the integration process.

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

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