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$$f(x) = \begin{cases} x^2 & x \leq 0 \\ \text{undefined} & x > 0\end{cases}$$

Will this function be differentiable at $x = 0$?

Because the left side limit is 0 and the right side limit doesn't exists, can one conclude that its derivative doesn't exists at $x = 0$?

Hanzel
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    It depends on who you ask. Some authors only define differentiability at points in the interior of the function's domain (which is not the case here). Others are more flexible, and will say the derivative is $0$. There's no one right answer to this. – Theo Bendit Oct 14 '22 at 11:57
  • To be clear, no function is allowed to be undefined on its domain. Your function is actually just $f(x)=x^2$ where the domain is specified to be $(-\infty,0]$. Your mention of "undefined" for $x>0$ is a poor choice for how to have conveyed that information. As for if it is differentiable there? Sure. When considering the limit as $x\to 0$ you need only consider values within the domain. The right side limit not only doesn't exist... it should not even be considered since no such values would matter as they are outside the domain of the function. – JMoravitz Oct 14 '22 at 11:57
  • Another interesting read on the topic: Most general $A\subseteq \Bbb R$ to define derivative of $f~:~A\to \Bbb R$? – JMoravitz Oct 14 '22 at 12:04

1 Answers1

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First of all, if we define a function $f:(-\infty,0]\to\mathbb R$ by $f(x)=x^2$, then it is not the case that $f(x)$ "equals" undefined when $x>0$. Rather, we would say that $f(x)$ is undefined when $x>0$, which simply means that $f$ is not defined at those points. The symbol $f(x)$ only makes sense in the context of when $x$ is a number in the domain of $f$, and outside of that context it has as much mathematical meaning as "-@4-sd:f" does.

Some authors require that for $f$ to be differentiable at a point $a$, it must be the case that $a$ is an interior point of the domain of $f$; that is, there must be an $r>0$ such that $f$ is defined on the interval $(a-r,a+r)$. Other authors define the derivative in a more general situation, e.g. if $a$ is a limit point of the domain of $f$.

According to the first definition of the derivative mentioned, $f$ is not differentiable at $0$. According to the second definition, $f$ is differentiable at $0$, and $f'(0)=0$.

Joe
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