(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose f:(a,b) to R(Real numbers), is differentiable at any point x that is in the interval (a,b). Prove that lim(as h goes to 0) of (f(x+h)- f(x-h))/(2h) exists and equals the derivative of f(x) ( which is f '(x)). Give an example of a function where the limit exists, but the function is not differentiable.

2. Relevant equations

The main equation that we can use is

f '(x)= the limit as h goes to 0 of (f(x+h)-f(x))/h (definition of the derivative)

Probably also need the fact that x is in the domain, and that x+h is in the domain.

3. The attempt at a solution

I played around with the lim (h goes to 0) (f(x+h)-(fx-h))/2h, and got

(1/2)((f(x+h)-f(x))/h) + (1/2)((f(x)-f(x-h))/h) but I am stuck on where to go next. Any ideas would be great. Thanks

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# Homework Help: Real Analysis Question involving the definition of the derivative

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