Proving a function is differetiable in R

[Simfish]

f(x) is a function with...

$$| f(x) - f(y) | \leq |x-y|^2 \forall x,y \in \Re$$

(a) prove differentiability in R, find f'
(b) prove f continuous

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my steps;

(a) $$\frac{| f(x) - f(y) |}{|x-y|} \leq |x-y|$$

Then by the definition of differentiability as stated in Apostol "Mathematica Analysis pg. 104, f is differentiable if the limit of the function as x -> y exists.

So by the inequality, as x -> y, we know that the limit is bounded and therefore must exist. The value of the limit is simply |x-y|, so is that always the derivative of the function? (or could it be |x|)? Since the derivative, after all, must always be positive?

(b) differentiability implies continuity

 

[SiddharthM]

Using epsilon-delta definition of limit, let epsilon=delta to show the limit of f(x)-f(y)/(x-y) is zero. that is as x goes to y.

derivatives aren't always positive. consider y=-x.

(-1)^n n a natural number is bounded but has no limit.