Suppose that $f(x)$ is continuous on $[0,1]$. We make an agreement that if there exists an interval $[a,b]\subseteq[0,1]$ including point $y$ such that $f(x)$ satisfies $\alpha$-Holder condition on $[a,b]$, $f(x)$ satisfies $\alpha$-Holder condition at $y$. Define $$A(y)=\sup\{\alpha : f(x) \text{ satisfies $\alpha$-Holder condition at $y$}\}.$$ What will the graph of $A(y)$ look like on $[0,1]$ for various $f(x)$, for example for functions of unbounded variation? Are there some constraints for $A(y)$?
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