Let the symmetric derivative of f at x be,
lim h->0 (f(x+h) + f(x-h) - 2f(x))/(h)
Assume f is continuous on the interval [0,1] and the symmetric derivative exists at all points in (0,1). Prove that there exists a point x in the open interval (0,1) where the ordinary derivative exists.
I've never seen anything like this before. Can anyone help?