Is there any example of a function $f$ on $R$ such that $$\left|\frac{f(x+h) + f(x-h) -2f(x)}{h}\right| \leq C$$ for all $x$ and all $h\neq 0 $ but not Lipschitz continuous(Lip$_1$) in $R$?
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Any discontinous additive function will work because it will have a. graph dense in $\mathbb{R}^2$ and the given quotient is $0$. Read about the construction here https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation
Ben Martin
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1@Sphere: In fact, this paper shows that there exist continuous functions satisfying this property (often called "Zygmund smooth" in the literature) that fail to be pointwise Lipschitz at each point. See also my comment to Prove the set of points where $f$ is differentiable is dense. – Dave L. Renfro Nov 30 '20 at 06:08