Let $f: [a,b] \to \mathbb{R}$ be a continuous functions. Let $\varepsilon>0$ be a constant. For $x \in [a+\varepsilon, b-\varepsilon]$, define $$g(x) : = \frac1{2\varepsilon} \int_{x-\varepsilon}^{x+\varepsilon} f(y)\ \mathsf dy.$$
$a)$ Show that $g$ is differentiable and find the derivative.
Attempt: Using FTC, I get $\frac{d}{dx} (\frac1{2\varepsilon}\int_{x-\varepsilon}^{x+\varepsilon} f) = \frac1{2\varepsilon} (f(x+\varepsilon) - f(x-\varepsilon))$.
$b)$ Let $f$ be differentiable and fix $x \in (a,b)$ (let $\varepsilon$ be small enough). What happens to $g'(x)$ as $\varepsilon$ gets smaller?
Attempt: $\lim_{\varepsilon \to 0} \frac1{2\varepsilon} (f(x+\varepsilon) - f(x-\varepsilon)) = \frac12 f'(x) = g'(x)$.
$c)$ Find $g$ for $f(x) : = |x|$, $\varepsilon = 1$ (You can assume $[a, b]$ is large enough).
Could you give some help for $c)$?