Let f : [a, b] → r be a continuous function. let ε > 0 be a constant. for x ∈ [a+ε,b−ε], define g(x) : = 1 2ε z x+ε x−ε f.
a) show that g is differentiable and find the derivative.
b) let f be differentiable and fix x ∈ (a, b) (let ε be small enough). what happens to g 0 (x) as ε gets smaller?
c) find g for f(x) : = |x|, ε = 1 (you can assume [a, b] is large enough).