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Let $f:\mathbb{R}\rightarrow \left [0,\infty \right )$ be a continuous function such that $g\left ( x \right )=(f\left ( x \right ))^2$  is uniformly continuous. Which of the following statements is always true ?

  1. $f$ is bounded
  2. $f$ may not be uniformly continuous
  3. $f$ is uniformly continuous
  4. $f$ is unbounded
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