The answer is option ©
For first option, let us suppose that $f$ is uniformly continuous. Then, we have
$||x|^{3/2}-|y|^{3/2}|$ $\leq$ |x-y|
Since the domain is $\mathbb R$, let us choose our $x$ and $y$ as $4$ and $9$ then left-hand side of the above inequation will be $|19|$ and the right-hand side will be $|5|$. Clearly, the above inequation does not hold for $x=4$ and $y=9$, and in fact you can find many such $x$ and $y$. Hence $f$ is not uniformly continuous.
For the second option, $f$ is definitely continuous and for that, you can apply the simple definition (assuming you know that), going to differentiability, $|x|^{a}$ is differentiable if $a>1$ because you will always get a smooth graph at $x=0$ and hence it will be differentiable at $x=0$., therefore this option will be incorrect.
Going to third option, we will have to find the derivative function of $f(x)$ which will be
$$Df(x)=\begin{cases}3/2x^{½}, &\text{if $x\geq 0$}\\-3/2x^{1/2}, &\text{if $x< 0$}\end{cases}$$
You can easily verify with the help of definition of continuity that this function is continuous at $x=0$. Hence third option is correct and fourth option is incorrect.