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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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.
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