- #1

Eclair_de_XII

- 1,083

- 91

## Homework Statement

"Suppose ##f:(a,b) \rightarrow ℝ## is differentiable at ##x\in (a,b)##. Prove that ##lim_{h \rightarrow 0}\frac{f(x+h)-f(x-h)}{2h}## exists and equals ##f'(x)##. Give an example of a function where this limit exists, but the function is not differentiable."

## Homework Equations

__Differentiability:__Let ##f:D\rightarrow ℝ## with ##x_0\in D'\cap D## (##D'## represents the set of ##D##'s accumulation points). For each ##t\in ℝ## such that ##x_0+t \in D## and ##t\neq 0##, define ##Q(t)=\frac{f(x_0+t)-f(x_0)}{t}##. The function ##f## is said to be differentiable at ##x_0## iff ##Q## has a limit at zero.

## The Attempt at a Solution

I know it's not really anything at all, but this is what I came up with for the first part:

##lim_{h\rightarrow 0} \frac{f(x+h)-f(x-h)}{2h}=lim_{h\rightarrow 0} \frac{f((x-h)+2h)-f(x-h)}{2h}=f'(x-h)##