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#1 2013-05-24 15:40:40

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Piecewise function

Can piecewise functions be differentiated?for example f(x)=2x for x<2,4 for x=2,x^2 for x>2,can f be differentiated?


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#2 2013-05-24 18:47:30

bob bundy
Moderator
Registered: 2010-06-20
Posts: 6,393

Re: Piecewise function

hi {7/3}

At points where the function is 'well-behaved', yes.  eg. For your example, everywhere except x = 2.

At this point, it may still be ok.  It is continuous there, so it passes that hurdle.  You then need to consider whether there is a left limit for the chord gradient and a right limit, and whether they are the same.

So, whilst f(x) = 2x, the gradient function is 2, for all x.

Whilst f(x) = x^2, the gradient function is 2x, so the right limit, as x tends to 2, is 4.

So there is not a consistency between the gradient as x approaches 2 from the left, and the gradient as x approaches 2 from the right.  So it is said that it is not differentiable at this point.

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2013-05-24 19:17:02

{7/3}
Member
Registered: 2013-02-11
Posts: 210

Re: Piecewise function

Thanks


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