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 $R$?
2026-04-01 08:08:00.1775030880
Difference quotients and Lipschitz
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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