Proving a function is constant, under certain conditions? [duplicate]

For all $x,t\in \Bbb R$ such that $x\neq t$, the equivalence below holds:

$$|f(t) - f(x)| \leq |t - x|^2\iff \left\vert \dfrac{f(t)-f(x)}{t-x}\right\vert\leq |t-x|,$$

taking the limit as $t$ approaches $x$ yields $$\lim \limits_{t\to x}\left(\left\vert \dfrac{f(t)-f(x)}{t-x}\right\vert\right)\leq \lim \limits_{t\to x}|t-x|=0.$$

This proves that $f$ is $\bbox[5px,border:2px solid #FFFFFF]{\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$ and $\forall x\in \Bbb R(f'(x)=\bbox[5px,border:2px solid #FFFFFF]{\_})$, thus $f$ is $\bbox[5px,border:2px solid #FFFFFF]{\_\_\_\_\_\_\_\_\_\_\_}$.

From the given expression We Can Write $\lim_{t\to x}-|t-x|\le \lim_{t\to x}{f(t) - f(x)\over t-x} \leq \lim_{t\to x} |t - x|$, now can you conclude $f'(x)=0\forall x$?