If $g(x):=f(x^2)$ is uniformly continuous, is $f$ a constant (assuming it's periodic)?

Question

I know that $f:\mathbb{R}\to\mathbb{R}$ is continuous and periodic. How can I prove that $f$ is a constant whenever $g(x):=f(x^2)$ is uniformly continuous?

Answer 1

Hint: Suppose $f$ is not constant (but still periodic, say, with period $p$). This means that there are $a$ and $b$ with $f(a)\ne f(b)$.

Now we're looking to prove that $g$ cannot be uniformly continuous.

In the definition of uniform continuity, set $\varepsilon = \frac12 |f(a)-f(b)|$. Can you see why there can be no positive $\delta$ that works?