$f(x) = \ln (x)$ and $f''(x) = -\frac{1}{x^2}$.
However, when $x=0$, $f''(0)=-\infty$.
Does $f(x)$ twice differentiable?
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7$f$ itself is not even defined at $x=0$, so it can't be differentiable there. – Mark Feb 28 '20 at 22:44
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5Note that $f(x) = \ln x$ is not even defined at $x=0$. However it is differentiable twice (and any number of times) at values $x$ where the logarithm is defined. – hardmath Feb 28 '20 at 22:46
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1thanks @hardmath. So, the differentiability of a function has a relationship to the domain of that function. thanks. – stackname Feb 28 '20 at 22:50
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4It is not defined at $x=0$, so you would not say that it is differentiable or it is not differentiable at that point. An old professor of mine said that it's like asking whether or not $\ln x$ is differentiable at the San Diego Zoo :) – Jair Taylor Feb 28 '20 at 22:59
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2It’s a fault of mathematics instruction in US high-schools that the importance of the domain of a function is not emphasized at all, sometimes not even mentioned. We can’t speak of differentiability of a function where it’s not defined, and we can’t speak of continuity of a function where it’s not defined, either. This leads to the lie that almost all high-school texts push, that the reciprocal function $f(x)=1/x$ is discontinuous at zero. It’s true that $0$ is a singularity of this function, but not a discontinuity, because the function is not defined at $0$. – Lubin Feb 28 '20 at 23:34
