It is known that if a real continuous function $f(x)$ satisfies a local $\alpha$-Hölder condition on a closed interval $[a,b]$, the box dimension of the graph of $f(x)$ on $[a,b]$ will be not greater than $2-\alpha$.

But if the proposition above is taken inversely, that is, if the box dimension of the graph of a real continuous function $f(x)$ on $[a,b]$ will be not greater than $2-\alpha$, how much can we guarantee that $f(x)$ satisfies a local $\alpha-$Hölder condition on a closed interval $[a,b]$?

For instance, the box dimension of the graph of $\sqrt{x}$ is one on $[0,1]$, and $\sqrt{x}$ satisfies a local $1$-Hölder condition on $[a,1]$ given any small positive $a$. Here we may say that $\sqrt{x}$ does not satisfy a local $1$-Hölder condition only at $0$, and one point of exception may not affect the value of the box dimension. So how much is such exception allowed that it does not affect the value of the box dimension?