= Solution
For $0<\alpha\le1$, uniform Lipschitz-$\alpha$ means that one constant $M$ satisfies $|f(x)-f(y)|\le M|x-y|^\alpha$ for every $x,y\in[0,1]$. The resulting <Hölder space> has <norm>
$$
\|f\|_{C^{0,\alpha}}=\|f\|_\infty+[f]_{C^{0,\alpha}},\qquad[f]_{C^{0,\alpha}}=\sup_{x\ne y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}.
$$
For the subsequent estimates at general $\alpha>0$, use the higher-order <Hölder class> convention: write $\alpha=r+\beta$ with $r\ge0$ an <integer> and $0<\beta\le1$, and require $f\in C^r([0,1])$ with $f^{(r)}$ uniformly Lipschitz-$\beta$. Thus
$$
\boxed{C^\alpha=C^{r,\beta},\qquad\|f\|_{C^\alpha}=\sum_{k=0}^r\|f^{(k)}\|_\infty+[f^{(r)}]_{C^{0,\beta}}.}
$$
At <integer> $\alpha=k$, this uses $C^{k-1,1}$; the alternative classical $C^k$ convention is stronger on this <compact> <closed interval> and also suffices for the estimates.
\b[For exponents above one, a first-difference inequality alone forces the <function> to be constant.] Partition $[x,y]$ into $n$ equal pieces and use the <triangle inequality>: $|f(y)-f(x)|\le M|y-x|^\alpha n^{1-\alpha}\to0$. Consequently a nontrivial higher-order definition is needed for the full range of exponents in the later <wavelet> estimates. Equivalently, higher-order <Hölder class> gives a local <Taylor polynomial> with remainder bounded by $C\|f\|_{C^\alpha}|x-x_0|^\alpha$.
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