Are $f'(a^+)$ and $f'(a^-)$ the notations for RHD and LHD?

So Is RHD given by $$\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$ mean $f'(a+)$ ? and in the same way LHD given by $$\lim_{h \to 0} \frac{f(a)-f(a-h)}{h}$$ mean $f'(a^-)$?

I got the doubt while studying the behavior of the function.

$$f(x) = \begin{cases} x^2 \sin(1/x) &\mbox{if } x \neq 0 \\ 0 & \mbox{if } x=0. \end{cases}$$

Since this function is differentiable at $x=0$ as RHD and LHD both are equal to zero i.e., $f'(0^+)=f'(0^-)=f'(0)=0$.

But as far continuity of $f'(x)$ is concerned we have

$$ f'(x) = \begin{cases} 2 x \sin \left(\frac{1}{x}\right)-\cos \left(\frac{1}{x}\right)&\mbox{if } x \neq 0 \\ 0 & \mbox{if } x=0, \end{cases}$$

here $f'(0^+)$ and $f'(0^-)$ are not defined

  • $\begingroup$ First of all, what's the question? And possibly related to standard notation for one-sided partial derivatives. $\endgroup$ – Simply Beautiful Art Jul 17 '17 at 17:10
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    $\begingroup$ No, it is not standard notation for $\lim_{h\to 0^+}\frac{f(a+h)- f(a)}{h}$. It is, however, standard notation for the limit as x goes to a of f'(x) which is what is intended here. $\endgroup$ – user247327 Jul 17 '17 at 17:19
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    $\begingroup$ Your edited question is still having notation issue. The notation for left hand right derivatives of $f$ is different from the left and right limits of the derivative $f'$. The notation $f'(a^{+}) $ represents right hand limit of $f'$ at $a$ and $f'_{+} (a) $ denotes right hand derivative of $f $ at $a$. Also note the basic result that if $f'(a^{+}) $ exists then $f'_{+} (a) $ also exists and is equal to $f'(a^{+}) $ but the converse does not hold. Similar notation and remarks apply to left hand derivative of a function and left hand limit of derivative of a function. $\endgroup$ – Paramanand Singh Jul 17 '17 at 17:43
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    $\begingroup$ Also both your limit definitions must use $h\to 0^{+}$ instead of usual $h\to 0$. $\endgroup$ – Paramanand Singh Jul 17 '17 at 17:51
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    $\begingroup$ $h$ is just a symbol for a variable. If you write $h\to 0$ then you can't say anything about sign of $h$. To emphasize sign of $h$ you need to write either $h\to 0^{+}$ or $h\to 0^{-}$. Also your line of thought is common in students and the typical belief is that $a-h<a<a+h$. Perhaps people tend to think that symbols are positive unless prefixed by a negative sign. One generally overcomes such beliefs with experience. $\endgroup$ – Paramanand Singh Jul 17 '17 at 17:57

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