I want to find a simple proof for, for $0<\alpha<1$, if $f\in C_c^0(\mathbb R^n)$ satisfies $|f(x+h)+f(x-h)-2f(x)|\le C|x|^\alpha$, then $f\in C^\alpha(\mathbb R^n)$.
I saw in Zygmund's original paper had this statement (lemma 4), also in Triebel's Theory of Function Spaces II. But Triebel doesn't give a direct proof as well.