= Hölder-Taylor remainder bound
{c}
{title2=$|f(x)-T_{x_0}(x)|\le C\|f\|_{C^\alpha}|x-x_0|^\alpha$}
Write $\alpha=r+\beta$ with an <integer> $r\ge0$ and $0<\beta\le1$, using $r=\alpha-1$ at <integer> $\alpha$. For $f\in C^{r,\beta}$, subtract its degree-$r$ <Taylor polynomial> at $x_0$. For $r\ge1$, the <integral> <Taylor remainder> with $f^{(r)}(t)-f^{(r)}(x_0)$ is bounded using the <Hölder seminorm>; for $r=0$ this is exactly <Hölder continuity>. The resulting bound is the displayed estimate. At <integer> order, the convention is $C^{r,1}$; a $C^{r+1}$ <function> on a <compact> <closed interval> also obeys this estimate.
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