Suppose that $f$ is a real-valued function which is twice differentiable in the interval $(-1,1)$. Does the following hold?:

$$\lim_{h \to 0} \frac{f(h) - 2f(0) + f(-h)}{h^2} = f''(0)$$

If $f''(x)$ is continuous this follows from Taylor's theorem with the Lagrange form of the remainder. What if $f''(x)$ is not continuous? -- Thanks.

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    $\begingroup$ I don't understand @CarloBeenakker's comment: of course it's perfectly possible for $f$ to be twice differentiable but for $f''$ to be discontinuous. Nevertheless, the question isn't appropriate for this forum. I'd suggest asking on math.stackexchange.com $\endgroup$ Mar 20, 2017 at 23:39
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    $\begingroup$ Hey guys, please check whether this question is nontrivial or not before deciding to shut it down. I may make a meta post in the reopening thread. (BTW, it seems the poster is in fact a professional mathematician.) $\endgroup$
    – Todd Trimble
    Mar 22, 2017 at 20:16
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    $\begingroup$ I think this follows directly from the "Peano form of remainder" appearing here on Wikipedia, which does not require continuity of the $k$th derivative. $\endgroup$ Mar 22, 2017 at 20:57
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    $\begingroup$ While I can see the possible rationale for deletion, it would be nice if some of the commenters above or those voting for deletion had a look at ams.org/mathscinet-getitem?mr=218905 $\endgroup$
    – Yemon Choi
    Mar 25, 2017 at 4:47
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    $\begingroup$ I see that I was wrong in voting to close, I have voted to reopen. I clearly misjudged the depth of the question (partly influenced, I have to confess, by the poor formatting and absence of motivating explanation). $\endgroup$
    – RP_
    Mar 25, 2017 at 17:50

1 Answer 1


Just so an answer is put here rather than in the comments:

As noted in the original question, the desired limit formula is a well-known, or at least straightforward, consequence of Taylor's theorem with Lagrange remainder in cases where $f\in C^2$, i.e. if we have some continuity of the 2nd derivative in a neighbourhood of $0$.

As pointed out by Nate Eldredge in the comments: for functions which are merely twice differentiable at the origin, i.e. where the second derivative exists at $0$ but might not be continuous in a neighbourhood of $0$, one still has Taylor's theorem (of order $2$) with Peano remainder: $$ f(x) = f(0) + f'(0)x + \frac{1}{2}f''(0)x^2 + h(x)x^2 $$ where $h(x)\to 0$ as $x\to 0$. This is enough to obtain the desired limit formula.

Remark. I took the liberty of contacting the OP and he said:

I am teaching the first real analysis course for math majors. In class I proved that if you know the value of a twice differentiable function at three points you know the value of the second derivative at some point ... and I assumed you need continuity of the second derivative to prove the limit formula.


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