设$f(x)$在$[a,b]$上有定义,$M>0$且对任意的$x$,$y\in[a,b]$,有$f(x)-f(y)|≤M|x-y|^k$
(1)证明:当$k>0$时,$f(x)$在$[a,b]$上连续;
(2)证明:当$k>1$时,$f(x) ≡$常数.
$\text{A.}$ 若 $\lim \limits _{x \rightarrow 0^{ }}f(x)=0$, 则 $\lim \limits _{x \rightarrow 0^{ }}f'(x)=0$
$\text{B.}$ 若 $\lim \limits _{x \rightarrow 0^{ }}f'(x)=0$, 则 $\lim \limits _{x \rightarrow 0^{ }}f(x)=0$
$\text{C.}$ 若 $\lim \limits _{x \rightarrow \infty }f(x)= \infty $, 则 $\lim \limits _{x \rightarrow \infty }f'(x)= \infty$
$\text{D.}$ 若 $\lim \limits {x \rightarrow \infty }f'(x)=A>0$, 则 $\lim \limits {x \rightarrow \infty }f(x)= \infty$