Prove that the Rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is uniformly continuous

Question

I need some help with a calculus homework question. Here is said question:

Let there be two polynomials $q$ and $p$ such that $\deg(p)\leq\deg(q)+1$ and $q(x)\neq0$ for all $x\in\mathbb{R}$.

Show that that rational function $f:\mathbb{R\rightarrow R}$ that is defined as $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ is uniformly continuous

I'm guessing that they want me to prove that all Rational Functions are uniformly continuous but I have no clue as to how to do that (seeing as throughout my internet search I saw many people referencing it)

Any help is appreciated, Thx!

Answer 1

Some hints:

• As the degree of $p(x)$ is smaller than the degree of $q(x)$ plus one, we have that $p(x)/q(x)$ behaves as $ax+b$ asymptotically.

• Now consider the function $h(x)=\frac{p(x)}{q(x)}-(ax+b)$, and note that $h(x)$ tends to zero as $x$ tends to plus/minus $\infty$.

• A continuous function is uniformly continuous on compact sets.

• The sum of two uniformly continuous functions is uniformly continuous

Do you see how to combine these statements into a proof? (Involves some case-checking).

Answer 2

In fact, such a function is Lipschitz continous but the degree assumption is critical here. This can be seen as follows: Let $n = \deg(p)$, $m = \deg (q)$. Since $q(x) \neq 0$, the function $f$ is differentiable for all $x \in \mathbb{R}$. Now, $$ f'(x) = \frac{p'(x)q(x) - p(x)q'(x)}{q(x)^2} $$ is bounded by the degree assumption (the denominator is of degree $2m$ while the numerator is of degree $\max\{(n-1) + m, n + (m-1)\} = n + m - 1 \leq 2m$). Thus $f$ is Lipschitz and in particular uniformly continuous.