Is there a bounded rational function that is not uniformly continuous on $\mathbb{R}$?

Question

I think the answer is no. I tried looking for one hour and it didn't work. Isn't every rational bounded function uniformly continuous ? There's nothing online about them.

Answer 1

Suppose $f=p/q$ is a rational function with coefficients in $\mathbb{R}$, written as a quotient of polynomials $p$ and $q$ in lowest terms. Then $f$ is bounded on $\mathbb{R}$ iff $q$ has no real roots (so $f$ has no real poles) and $\deg p\leq \deg q$ (so $f(x)$ stays bounded as $x\to\pm\infty$).

Now suppose $f$ is bounded and consider the derivative $$f'=\frac{p'q-pq'}{q^2}.$$ We see that the denominator of $f'$ also has no real roots and the degree of the denominator is at least that of the numerator (in fact, the degree of the denominator is now strictly greater). Thus $f'$ is also bounded on $\mathbb{R}$. It follows that $f$ is Lipschitz and in particular uniformly continuous.