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I read that the derivative of a function can never have a "jump" discontinuity, but only essential discontinuity. My question is, can the derivative have a "half essential and half jump" discontinuity, where $\lim_{x\to a^-}f(x)$ does NOT exist, $\lim_{x\to a^+}f(x)$ DOES exist, and $\lim_{x\to a^+}f(x)\not= f(a)$ ?

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  • $\begingroup$ I suppose you should write $f'(x)$ in place of $f(x)$ in your question, because you are concerned about the discontinuity of the derivative $f'(x)$ and not $f(x)$. $\endgroup$
    – Paramanand Singh
    Mar 15 '16 at 4:06
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The precise meaning of "derivatives can't have jump discontinuity" is the following:

Let $$f'_{-}(a) = \lim_{h \to 0^{-}}\frac{f(a + h) - f(a)}{h},\,f'_{+}(a) = \lim_{h \to 0^{+}}\frac{f(a + h) - f(a)}{h}$$ If $\lim_{x \to a^{+}}f'(x)$ exists then $f'_{+}(a)$ also exists and $$f'_{+}(a) = \lim_{x \to a^{+}}f'(x)$$ and if $\lim_{x \to a^{-}}f'(x)$ exists then $f'_{-}(a)$ also exists and $$f'_{-}(a) = \lim_{x \to a^{-}}f'(x)$$

Thus there are two numbers related to the behavior of $f$ on the left side of $x = a$ and two numbers related to the behavior of $f$ on the right side of $x = a$. Numbers on each side have to be equal if they exist. There can be differences if you compare the numbers on one side to numbers on the other side. Andre Nicolas's answer gives an example of this behavior.

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