Consider the functions $f_{1}, f_{2}:(0, \infty) \rightarrow \mathbb{R}$ defined by
\[
f_{1}(x)=\sqrt{x}, \quad \text { and } \quad f_{2}(x)=\sqrt{x} \sin x .
\]
Which of the following statements is correct?
- $f_{1}$ and $f_{2}$ are uniformly continuous.
- $f_{1}$ is uniformly continuous, but $f_{2}$ is not.
- $f_{2}$ is uniformly continuous, but $f_{1}$ is not.
- Neither $f_{1}$ nor $f_{2}$ is uniformly continuous.