Analysis of the Differential-Difference Equation $y(x+1/2)-y(x-1/2) = y'(x)$

In this paper we study some solution techniques of differential-difference equation $$y'(x) = y(x + 1/2)- y(x- 1/2),$$ first without an initial condition and then with some initial function $h$ defined on the unit interval $[-1/2, 1/2]$. We show some sufficient conditions that an initial function $h$ is admissible, i.e., it yields a unique continuous solution on some symmetric interval about $0$.