Math Is Fun Forum

Dharshi

Member

Registered: 2006-10-31

Posts: 56

Need helpppppppppppp!!!!!!!!!!!!!!

Give an example of a function which is continuous on the whole real line but fails to be differentiable at exactly 2 points.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Need helpppppppppppp!!!!!!!!!!!!!!

This isn't too bad of a question.  Just remember one thing: You can define a function piecewise:

f(x) =

x^2 if x > 0
-x if x <= 0

This function is continuous.  It is definitely differentiable at everywhere but 0.  Is it differentiable at 0?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Need helpppppppppppp!!!!!!!!!!!!!!

Another example:

This one is continuous everywhere and not differentiable at exactly two points (which is what is required).

Dharshi

Member

Registered: 2006-10-31

Posts: 56

Re: Need helpppppppppppp!!!!!!!!!!!!!!

Another example:

This one is continuous everywhere and not differentiable at exactly two points (which is what is required).

Oh!! Thanks a lot.:lol:

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Need helpppppppppppp!!!!!!!!!!!!!!

You’re welcome. The important question though is: do you know why it is not differentiable at those two points where it’s not differentiable?

Dharshi

Member

Registered: 2006-10-31

Posts: 56

Re: Need helpppppppppppp!!!!!!!!!!!!!!

You’re welcome. The important question though is: do you know why it is not differentiable at those two points where it’s not differentiable?

I think the graph has holes on (-1,1).
Am I right?

HallsofIvy

Guest

Re: Need helpppppppppppp!!!!!!!!!!!!!!

You’re welcome. The important question though is: do you know why it is not differentiable at those two points where it’s not differentiable?

I think the graph has holes on (-1,1).
Am I right?

No, you are not.  The graph having "holes" would mean the function is not continuous.   That is not the case here.  Have you drawn the graph?  I recommend you do that.

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

Re: Need helpppppppppppp!!!!!!!!!!!!!!

Here's another one with absolute value.  Enjoy!
      y = | 6 - x^2 |
First figure it out without the absolute value signs, then anything below the
x-axis, bounce it up to the positive area above the x-axis.

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