Uniform continuity of $f(x) = \frac{2x^5-98}{(x-1)(x-9)}$ on $(2, 8)$

Question

Problem : Show that the function $f(x) = \dfrac{2x^5-98}{(x-1)(x-9)}$ is uniformly continuous on $(2,8)$.

I've tried doing this by a direct proof and using the definition to try and factor out a '$(x-y)$' so as to use the triangle inequality but I got stuck halfway trying to simplify the function.

I've also tried using Lipshitz functions to prove it but I'm not sure if it works on an open interval.

Any help for this please?

Answer 1

Hint:

Prove that the function is uniformly continuous on $[1.5, 8.5]$. Note that it is a superset of $(2,8)$. Use the definition of uniformly continuous to make the conclusion.

Answer 2

$f$ is continuous on the compact intervall$[2,8]$, hence uniformly continuous on the compact intervall $[2,8]$, hence uniformly continuous on the open intervall$(2,8)$.

Answer 3

Using partial fraction decomposition, you can write $f(x)$ as the finite ( about $6$ terms at most ) sum of rational functions of the form $\dfrac{p(x)}{x-1}$ and $\dfrac{q(x)}{x-9}$ whereas $p,q$ are polynomials of degree $4$ or lower. Thus you can prove these functions are uniformly continuous with $\epsilon$ - $\delta$ argument and invoke the property that the finite sum of uniformly continuous functions is again a uniformly continuous function over the given interval. I leave the details for you to fill out.

Answer 4

You can also do it using continuous extension theorem. As this function is continuous at 2 and 8 and also (2,8). Hence it is uniformly continuous on (2,8)