Continuous function with uniformly continuous square.

Question

Let $f:\mathbb{R} \rightarrow [0,\infty)$ be a continuous function such that $g(x) = (f(x))^2$ is uniformly continuous. Prove or disprove that $f$ is uniformly continuous.

According to me $f$ should be uniformly continuous but I am facing difficulty in proving that.

Thanks in advance!

Answer 1

Note that $f = h \circ g$, where $h:[0,\infty)\rightarrow[0,\infty)$. Now:

• $g$ is uniformly continuous (the assumption).
• $h$ is uniformly continuous ($\sqrt x$ is uniformly continuous).

• The composition of two uniformly continuous functions is uniformly continuous (composition of two uniformly continuous functions.).

  It follows that $f$ is uniformly continuous.