Math Is Fun Forum

{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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Bob

Administrator

Registered: 2010-06-20

Posts: 10,143

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

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
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{7/3}

Member

Registered: 2013-02-11

Posts: 210

Re: Piecewise function

Thanks

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There are 10 kinds of people in the world,people who understand binary and people who don't.