Let f be a continuous function on open interval I.
Let h>0 and a<b so that: (a-h,b+h) c I.
Now we choose d: 0 < d< h
and now for every x in(a,b) we define the following function:
G(x) = 1/2d * {int(-d --> d) f(x+t)dt}
Prove:
1. G is differentiable functions whose derivative is continuous.
2. lim(d-->0+) G(x) = f(x)
Thank you for reading!