Suppose f(x) is differentiable at ALL x in R.
Is it possible for lim x->0+ f '(x), and lim x->0- f '(x) to exist and NOT be equal?© BrainMass Inc. brainmass.com September 23, 2018, 4:18 am ad1c9bdddf - https://brainmass.com/math/discrete-math/lim-lim-exist-differ-174873
The solution is attached.
Since f is differentiable everywhere, exists.
Then, using the Mean Value Theorem,
We are assuming that exists. That is, given any , there exists ...
A differentiable function is continuous at any point for which the limit of the derivative exists. The solution is a step by step proof of that fact comprising 3/4 of a page in Word with equations written in Mathtype. (Although the question is not worded in that way, that is in fact what is being proved) The proof uses the mean value theorem, which is frequently useful in such proofs and so serves as a useful illustration. Also given is an example of a function with a discontinuous derivative.