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Calculus/Differentiability

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Expert: - 7/22/2010


Question
Hello Sir
I was confused by this problem;

I know the function |X-1|(absolute Value) is not differentiable at one because it has sharp corner in that point
but is the function differentiable in (-∞,1] including one
and on [1,∞) including one?
I am confused since it is not differentiable at 1 how can those inclusive braces be used?

Answer
I don't think they should be used, sinde the function is not differentiable at that point.

It can be said to be differentiable on  (-∞,1) and  (1,∞).

To be differentiable at a point, the function must have the same slope on both sides of that point, and this function has slope 1 on the right sides and slope -1 on the left side,
so I don't see why the put in ...,1] or [1,... when it should be ...,1) and (1,...

It can be said that the function is coninuous on (-∞,1] and [1,∞),
but it can also be said it is continuous on (-∞,∞) since it is coninuous up to 1 on both sides.

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