L'hopital's Rule and f''(a) Existence

[Kate2010]

Homework Statement

I have to show that lim_h->0 [(f(a+h)-2f(a)+f(a-h))/h²] = f''(a) where f:R->R is differentiable, a is a real number and f''(a) exists.

Homework Equations

The Attempt at a Solution

I have applied l'hopitals rule as the question advises and have got to f''(a). However, my problem lies in checking the conditions for l'hopitals rule each time. I'm ok with showing the f(x) (numerator) and g(x) are zero when x=0 for each case. However, I am unsure which region I can take for differentiable and continuous. I thought maybe differentiable on (-2a,2a) \ {0} and continuous on [-2a,2a]. I have used these regions each time. But I am also unsure if I can claim that as f''(a) exists then f' is differentiable in the region (-2a,2a)/{0} and continuous in [-2a,2a]. If f''(a) exists, f' is not necessarily differentiable in this region? Or is it? I wasn't sure what else I could do.

 

[Kate2010]

Or would it be better to say continuous in the region [a-2h,a+2h], differentiable in the region (a-2h,a+2h)\{0}. Is this allowed even though h is tending to 0?