Purchase Solution

Can lim x->0+ f '(x) and lim x->0- f '(x) exist and differ?

Not what you're looking for?

Ask Custom Question

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?

Purchase this Solution

Solution Summary

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.

Solution Preview

The solution is attached.

Since f is differentiable everywhere, exists.

Then, using the Mean Value Theorem,
for some

We are assuming that exists. That is, given any , there exists ...

Purchase this Solution


Free BrainMass Quizzes
Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.