Define $g(x)$ as follows: \[ g(x)= \begin{cases} -1, & \text{if } x < 0,\\ 0, & \text{if } x = 0, \\ 1,

$f(x)\ =\ g(x+1)-g(x-1)\\~\\ \color{purple}{f(\frac12)}\ =\ g(\frac12+1)-g(\frac12-1)\\~\\ \color{purple}\phantom{f(\frac12)}\ =\ g(\frac32)-g(-\frac12)\\~\\ \color{purple}\phantom{f(\frac12)}\ =\ 1--1\\~\\ \color{purple}\phantom{f(\frac12)}\ =\ 2\\~\\ \color{Green}{f(-\frac12)}\ =\ g(-\frac12+1)-g(-\frac12-1)\\~\\ \color{Green}\phantom{f(\frac-12)}\ =\ g(\frac12)-g(-\frac32)\\~\\ \color{Green}\phantom{f(\frac-12)}\ =\ 1--1\\~\\ \color{Green}\phantom{f(\frac-12)}\ =\ 2\\~\\ f(\frac12)+f(-\frac12)\ =\ 2+2\ =\ 4$

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