Existence of the derivative at a point is implied by a version of the symmetric derivative plus continuity This is problem 1.1.1.(ii) on p.10 from Flett's book "Differential Analysis". Variants of the problem have appeared in this forum under the subject of symmetric derivative (e.g., here and here). Flett phrases the problem in terms of functions having a normed space as codomain, and limits, but I am having a really hard time figuring out the simplest case of a function $f:\mathbb R\to\mathbb R$ such that


*

*for all sequences $(x_{n})$ and $(y_{n})$ satisfying $x_{n}>c>y_{n}$ and converging to $c$ the limit $\lim_{n\to\infty}\frac{f(x_{n})-f(y_{n})}{x_{n}-y_{n}}$ exists and equals $L$;

*the function $f$ is continuous at $c$.


Then, the derivative of $f$ at $c$ exists, and moreover $f^{\prime}(c) = L$.
A (sort of) converse to this result states that if $f$ is differentiable at $c$, then its symmetric derivative, meaning the limit $\lim_{h\to 0}\frac{f(c+h)-f(c-h)}{2h}$, exists and equals $L$. This is not difficult to prove using a middle-man trick. 
What I cannot figure out is how the additional condition that $f$ is continuous at $c$ on top of (a version of) the symmetric derivative at $c$ guarantees that $f$ is differentiable at $c$ and $f^{\prime}(c)$ equals the symmetric derivative at that point. Any suggestions?
 A: What follows are results that specifically relate to the ordinary differentiability of continuous functions having everywhere a (finite) symmetric derivative.
In [1], after observing that it is known that a continuous function which is symmetrically differentiable everywhere must have an ordinary derivative everywhere except for a set that is both Lebesgue measure zero and first (Baire) category, and that it is not difficult to find examples where this "small" exceptional set is countably infinite, Foran showed there exists a function continuous everywhere and symmetrically differentiable everywhere such that it fails to have an ordinary derivative at each point of a certain perfect nowhere dense set (i.e. a set of cardinality $c).$ Foran notes that the exceptional set for his example has Hausdorff dimension $0,$ and he posed the question of whether such an example exists in which the exceptional set has positive Hausdorff dimension.
In [2], Belna/Evans/Humke proved that the exceptional set is upper $\sigma$-porous, a notion implying both Lebesgue measure zero and first category, but not conversely. Although it was known that there exist sets of positive Hausdorff dimension that are $\sigma$-porous (even sets having Hausdorff dimension $1$ in every interval), it was still not known whether the exceptional set in this symmetric/ordinary derivative theorem for continuous functions could have positive Hausdorff dimension. The size of the exceptional set was strengthened to the very refined notion of a $\sigma-(1-\epsilon)$-upper symmetrically porous set in [3], but again nothing about Hausdorff dimension was yet known, and there exist sets having this strictly stronger notion of smallness that have Hausdorff dimension $1$ in every interval. (For more about this "very refined notion" and the results I've stated, see this paper.)
Finally, in [4] Zajíček answered positively the question of whether the exceptional set can have positive Hausdorff dimension by a proving that any countable union of sets, each of which is closed and $1$-symmetrically porous, can be the set at which an everywhere (finitely) symmetrically differentiable Lipschitz continuous function can fail to have an ordinary derivative. Since there exist closed $1$-symmetrically porous sets having Hausdorff dimension $1,$ this result implies, among other things, that the exceptional set can have Hausdorff dimension $1$ in every interval (by putting a closed $1$-symmetrically porous set of Hausdorff dimension $1$ in each open interval with rational endpoints, and taking the union of these sets).
[1] James M. Foran, The symmetric and ordinary derivative, Real Analysis Exchange 2 #2 (1977), 105-108.
[2] Charles L. Belna, Michael J. Evans, and Paul D. Humke, Symmetric and ordinary differentiation, Proceedings of the American Mathematical Society 72 #2 (November 1978), 261-267.
[3] Luděk Zajíček, A note on the symmetric and ordinary derivative, Atti del Seminario Matematico e Fisico dell'Università di Modena 41 (1993), 263-267.
[4] Luděk Zajíček, Ordinary derivatives via symmetric derivatives and a Lipschitz condition via a symmetric Lipschitz condition, Real Analysis Exchange 23 #2 (1997-1998), 653-669.
A: Note: The following is maybe too specific to be called a hint... More like an outline of proof.
If $(h_n)_n$ satisfies $h_n>0$ for every $n$ and $h_n\to0$ as $n\to\infty$ then
$x_n=c+h_n$ and $y_n=c-h_n$ satisfy the given hypothesis.
We estimate the difference
$$
\left| \frac{f(c+h_n)-f(c)}{h_n} - \frac{f(x_n)-f(y_n)}{x_n-y_n} \right|
$$
using the continuity of $f$ at $c$.
Given $\varepsilon>0$ we then estimate the difference
$$
\left| \frac{f(c+h_n)-f(c)}{h_n}-L \right|
.
$$
This gives of $f'(c+)=L$, that is, $\lim_{h\to 0^+}(f(c+h)-f(c))/h=L$. The rest ($f'(c-)=L$) is analogous.
A: Hint: Let $x_n\to c^+.$ For any $n,$ we have
$$\frac{f(x_n)-f(c)}{x_n-c} = \lim_{m\to \infty}\frac{f(x_n)-f(c-1/m)}{x_n-(c-1/m)}.$$
Here we have used the continuity of $f$ at $c.$
