Let $x$ be real and uniformly continuous on a set $I$ and Hölder with exponent $\alpha$ and constant $a$ on a dense subset $A$ of $I$. Then $x$ is Hölder on $I$.
By density of $A$, there exists $(s_n),(t_n)$ with $s_n\to s$ and $t_n\to t$. As $x$ is uniformly continuous on I, we have \begin{align*} |x(s)-x(t)|&\leq |x(s)-x(s_n)|+|x(t)-x(t_n)|+|x(s_n)-x(t_n)| \\ &\leq |x(s)-x(s_n)|+|x(t)-x(t_n)|+a|s_n-t_n|^\alpha \\ &\to a|s-t|^\alpha ~(n\to \infty). \end{align*}
Why do we need uniform continuity and not just continuity?
This is not the same as the other question. One about extension of a continuous function and the other about compactness of set. Do you even read