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I have been trying to solve the following problem:

Suppose the function $f:\mathbb R \rightarrow \mathbb R$ has left and right derivatives at $0$.Then at $x=0$, which of the following options is correct?

(a)$f$ must be continuous but may not be differentiable,
(b)$f$ need not be continuous but must be left continuous or right continuous,
(c)$f$ must be differentiable,
(d)if $f$ is continuous then $f$ must be differentiable.

Could someone point me in the right direction.Thanks in advance for your time.

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1 Answer 1

The function $f(x)=|x|$ should guide you towards the right answer. Also, if $f(x)$ is left-differentiable, what can you say about the left limit?

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Sir,since $f$ is left-differentiable,it should be left-continuous.here the left limit for the function is $0$. –  user53386 Dec 17 '12 at 17:51
And since it is right differentiable, it is also right continuous ;) What can you say about a function which is both left and right continuous? –  N. S. Dec 17 '12 at 18:00
@N.S. is there no issue due to the fact that we dont know the left derivative is equal to the right derivative? –  MSEoris Dec 17 '12 at 18:03
@MSEoris Nope. $\lim_{x \to x_o^+} f(x)-f(x_0)=\lim_{x \to x_o^+} \frac{f(x)-f(x_0)}{x-x_0}(x-x_0)=f'^{+}(x_0) \cdot 0=0$... In the computation it is irrelevant what the left/right derivatives are, since they are multiplied by $0$. –  N. S. Dec 17 '12 at 18:07
I think the function is continuous as for the function being both left and right continuous and its value being equal to $0$. –  user53386 Dec 17 '12 at 18:11

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