Give an example of a function such that $f’(0)$ exists but $$\lim_{x \rightarrow 0}f(x)$$ does not exist.
Hello, I am struggling to find an example of this? Much help, thanks!
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What have you tried? – Dominic Petti Feb 21 '21 at 23:45
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3If $f'(0)$ exists, then $f(x)$ is continuous at $x=0,$ and hence $\lim\limits_{x \rightarrow 0}f(x)$ exists. Therefore, there is no such example. – Dave L. Renfro Feb 21 '21 at 23:50
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Assuming the common rigorous definition of the derivative, differentiability implies continuity, and for a function continuous in 0, we have that $\lim_{x\to0}f(x) = f(0)$, in particular the limit exists.
Sometimes at a discontinuity there can be a left and a right limit of the derivative, and when they are equal their common vale is taken as the derivative. An example would be a function that is 0 up to and including 0, and 1 for $x>0$.
doetoe
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1There is only one accepted definition of derivative. That definition is equivalent to the statement that the left and right limits defining the derivative exist and are equal and that the the function is continuous. Your example (the characteristic function of the interval $(0, \infty)$) does not have a derivative at $0$. – Rob Arthan Feb 21 '21 at 23:57
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You said that sometimes the common value of the left and right derivative is taken to be the derivative at a discontinuity. That does not agree with the accepted definition of the derivative. – Rob Arthan Feb 22 '21 at 00:12
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@RobArthan Yes, which I also say (assuming the common rigorous definition this is not possible, in the first paragraph). Which is why I suspect that OP's text does not follow the accepted definition, but maybe something like the other (which is actually stated our implied in some physics or engineering texts) – doetoe Feb 22 '21 at 00:31
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