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If $f(x)$ is a function for which $\lim \limits_{x \to b} \frac{f(x)-f(b)}{x-b}=0$, then which of the following is true?

A

$\lim \limits_{x \to b^+} f(x) \neq \lim \limits_ {x \to b^-} f(x)$

B

$f(b)=0$

C

$f(x)$ is differentiable at $x=0$

D

$\lim \limits_{x \to b} f(x)=f(b)$

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