Continuity and differentiability at a point w/ absolute-value function

[einstein007]

Hi,

Here is my question:

Is the function y = f(x) = x|x| continuous and differentiable at x = 0 ? I drew the graphic and it "seems" that the answer is yes to both questions.
Can someone help me please?

Thank you !

 

[einstein007]

As I can see, my question is too complicated for you... I will ask my professor.

 

[ksdhart2]

Wow. What a stunning amount of impatience. As noted in the Read Before Posting (you did read it, yes? ):

Have patience. There is no paid staff waiting on-hand to give instant replies. Many of the volunteer tutors have "real" jobs, and they all have to sleep from time to time. The people "viewing" your posts may be fellow students. Please don't be offended if there are "views" but no replies. It may take hours, even days, for a tutor, qualified in your topic's area, to respond.

I'm so terribly sorry that there wasn't a volunteer tutor here ready to just drop everything and answer your question within an hour!

Now, with that mini-rant out of the way, let's see about that question of yours, shall we? As you know, just visually inspecting the graph can prove neither continuity nor differentiability. Your best bet would be to return to the definitions of those terms. The definitions lay out several criteria for a given function to be continuous and/or differentiable at some point. What are those criteria? Does your given f(x) meet those criteria? Why or why not? How does that help you solve the problem?

If you need a refresher on these definitions, you might try here for continuity and here for differentiability.

 

[ksdhart2]

Because I'm a very patient man, I'm willing to give you one more chance. I asked several leading questions to point you in the right direction. Did you even try to answer these questions? That's how this forum works, by the way - you show us what you've tried and exactly what's giving you troubles, and we point out any errors you might have made and/or hints as to how to proceed next. If you return to the definitions, look carefully at what they say, and then answer the leading questions I asked you, you'll arrive at the answer.

 

[HallsofIvy]

A function is "continuous" at a point, x= a, if and only if \(\displaystyle \lim_{x\to a^-} f(x)= \lim_{x\to a^+} f(x)= f(a)\).

Further, while the derivative of a differentiable function is not necessarily continuous, it does satisfy the "intermediate value property" so that \(\displaystyle \lim_{x\to a^-}f'(x)= \lim_{x\to a^+} f'(x)\).

Is that true here?