Decide whether each of the following statements is true or false. If true, explain why. If false, give a counter-example and explain why the counter-example contradicts the statement.

Suppose F(x) is differentiable at ALL x in R.

Suppose lim x->0 f '(x) = L, does it follow that f '(0) = L?

Solution Summary

Differentiability and limits are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

Real Analysis : Continuity, Closed and Open Sets and Differentiability. ... is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s ...

... 0. We can analyze the differentiability of h at x = 0 by computing formally its derivative and checking the existence of the limit of derivative at x = 0: ( 9 ...

... Since f is differentiable on (a,b), f is differentiable at . BY definition, we have. ... Note. If exists, then the left limit and right limit must exist and equal. ...

... Hence ' Since ex has no limit as x → ∞ , we say that f ( x) doesn't tend to a finite limit A whenever x → ∞ . 3. Let f be differentiable for x > a and ...

... In any case, we will first prove that this function is once differentiable, with f'(0 ... we need to find the derivative at x = 0 by looking at the limit of the ...

... To show that f(x) is differentiable at x, you should prove that this limit exists: lim[f(x+h)-f(x)]/h ; (h->0) and then the result of this limit is f'(x ...

... 1. Proof: Since f is twice differentiable, according to the Mean Value Theorem, we can ... cn )} is a bounded increasing set of numbers and thus must have a limit. ...

... in two parts such that in each part we are dealing with a differentiable function. ... from epsilon to infinity with epsilon > 0 and then take the limit of epsilon ...

...limit point of M_K is also in M_K , therefore M_K is closed as defined on page 49. (aC) To prove that M_K is the closure of the set D_K of all differentiable...