If $f’’(a)$ is defined, is $f$ differentiable near $a$? My logic goes, if $f’’(a)$ exist, then $f’(a)$ must be defined near $a$, thus $f$ is differentiable near $a$. Is that correct?
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2Yes, this is correct. Differentiability in one dimension is usually introduced as a pointwise property but for functions which are defined on an interval. For $f''(a)$ to exist $f'$ has to exist on an interval containing $a$. – Gerd Mar 28 '24 at 12:41
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@Gerd, that depends on the definition of $f''$. It can be defined by a single limit, for example $f''(a)=\lim \frac{f(a+h)+f(a-h)-2f(a)}{h^2}$. – Mikhail Katz Mar 28 '24 at 12:43
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1@MikhailKatz Sure, but I guess that the question refers to the classical definition. Maybe this should be clarified by mafsu. – Gerd Mar 28 '24 at 12:47
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@geetha290krm Isn't the function you describe differentiable at $0$? – legionwhale Mar 28 '24 at 12:48
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@MikhailKatz That seems like quite a strange definition to me. By that definition, it would seem that $f''$ exists at $0$ for every odd function, even if $f$ is everywhere discontinuous? – legionwhale Mar 28 '24 at 12:55
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@legionwhale, this definition is quite useful in differential geometry when working with curvatures. It can be written in a more general way to avoid making every odd function possess $f''(0)$, if you prefer. The OP should clarify which definition he has in mind. – Mikhail Katz Mar 28 '24 at 13:11
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Well to expand a bit, this question came up when trying to solve a limit using lhospital. Specifically 2f(x)/x^2, x approaching 0, where f(0)=f’(0)=0 and f”(0)=1. My thought was if I could use lhospital rule without knowing if f is differentiable, since f”(0) is defined. I’m in high school so still learning this stuff – mafsu Mar 28 '24 at 13:15
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1If so, the intention of the problem is to assume that $f$ is twice diffentiable in a neighborhood of the point, and apply l'Hopital's rule twice. It doesn't seem that you are expected to worry about types of differentiability. – Mikhail Katz Mar 28 '24 at 14:17
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@mafsu - a lesson for you to be learned here: if you ask a question of mathematicians without being very specific about the context, they will leap off in highly unexpected directions far removed from your intent. You have to keep us on a short leash if you don't want us dragging you all around the dog park. As far you need to be concerned, the second derivative is the derivative of the derivative, which requires that $f'(x)$ exists in some neighborhood of $x = a$. – Paul Sinclair Mar 29 '24 at 16:17