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Does continuity implies differentiability?

So i just need to check for continuity at that point?

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Differentiability implies continuity, the other way around does not hold. The common counterexample is the function $x\mapsto |x|$. It is continuous at 0 but not differentiable there. –  Stefan Hansen Sep 29 '12 at 12:41
    
What condition(s) need to be true for a function to be differentiable at a point? This is a good place to start... –  Daryl Sep 29 '12 at 13:00

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up vote 3 down vote accepted

As Stefan says, continuity does not imply differentiability. So, your question is, how do you check differentiability. The answer is, use the limit definition. The definition says that $f'(x)$ exists if the following limit exists:

$$\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$

Check the example given by Stefan specifically for a good example. In that example, you will need to calculate the left and right hand limits separately and you will find that they are different. Therefore, the limit does not exist. Therefore, $|x|$ is not differentiable at 0.

One way that is not rigorous, but that might be helpful, is to know that derivatives won't exist at sharp points (because the left and right hand limits will be different, this is where $|x|$ fits) or at places of vertical tangents (because the limit is $\pm\infty$, i.e., it does not exist as a real number). An example of the latter would be the function $f(x) = x^{1/3}$ at the point $(0, 0)$. Again, go ahead and try calculating the limit definition of the derivative at 0 and see what you get.

Now, of course, you could use continuity in a small number of cases. Since differentiability implies continuity, not continuous implies not differentiable. So, if a function is not continuous at a point, you can know without any extra work that it is not differentiable there.

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Great! I loved your last point about countynity. –  Yellow Skies Sep 29 '12 at 13:14

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