Stein in his textbook mentioned that one can come up with a nowhere differentiable function that $$f(x+t) + f(x-t) - 2f(x) \leq C\vert t \vert \quad \text{for} \quad \vert t\vert \leq r(x)$$ at every point $x$. Here $C$ is independent of $x$ but $r(x)$ could vary with $x$. Sadly, he did not bother to explain how one could construct such a function. I have also tried standard example like Weierstrass function, but it seems not working.
Edit: Thanks to Jose27's suggestion, I found it in theorem 4.9 of chapter 2 in Zygmund's book. Funnily enough, the example given is exactly the Weierstrass functuon $$f_1(x)=\sum^{\infty}_{n=1} b^{-n}\cos(b^n x)$$ that I presumed was not working.