URGENT HELP ON: Differentiability and continuity Watch

lauzy
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#1
Report Thread starter 12 years ago
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Hey!

Is it possible to find a function which is differentiable at 0 and not continuous at 0?
And is it possible to find a function which is differentiable at 0 and not continuous at 1?

I know that differentiability implies continuity so it wouldn't be possiblt to be diff at 0 and cont at 0 but what about the second q? Does diff only imply continuity if its at the same point?

Thanks :confused:
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TheDuck
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For the second xsin(1/(x-1))


(edited)
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jpowell
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1) No, differentiability at a point implies continuity, if a function is not continuous at x0 then it is not differentiable at x0.

2) Of course, define a function that is a from -inf to 1 and b from 1 to inf a not equal to b.... that is differentiable at 0, and not continuous at 1.

Differentiability and continuity are local properties of functions, i.e. differentiability at x0 implies nothing about x1, and vice versa.
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Wrangler
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Obviously if a function is differentiable at a point then it must be continuous there. Your second question is strange - you want to know if there exist functions that are differentiable at zero and discontinuous at 1? Well, just take your favourite function that happens to be differentiable at 0 and insert a discontinuity at 1.

Do you perhaps want to know about functions that are differentiable at a point, that the derivative is discontinuous there?
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lauzy
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Oh I see so u could use for example the function that theduck suggested would be diff at 0 and not cont at 1, cool thanks guys
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Christo
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(Original post by Wrangler)
Obviously if a function is differentiable at a point then it must be continuous there. Your second question is strange - you want to know if there exist functions that are differentiable at zero and discontinuous at 1? Well, just take your favourite function that happens to be differentiable at 0 and insert a discontinuity at 1.

Do you perhaps want to know about functions that are differentiable at a point, that the derivative is discontinuous there?
Just like a lecturer! They're always telling me to take my favourite positive integer or somesuch thing..
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