Proof of uniform continuity of a function

Question

Show that the function $f(x) = \cfrac{x^2 + 5x - 7}{(x^2 - 9x + 8)(x-2)}$ is uniformly continuous on the interval $(3,5)$ (not with epsilon and delta)

How do I do this question? I am sitting an exam in an hour, and this is likely similar to one of the questions that is on it. Help please

Answer 1

The denominator does not vanish in the interval $[3,5]$. Thus our function is continuous on the closed bounded interval $[3,5]$, and therefore uniformly continuous in that interval. So it is uniformly continuous on $(3,5)$.

Answer 2

Remember that

• If $g$ is continuous and its domain is a closed interval $[a,b]$, then $f$ is uniformly continuous (it's a theorem that can be found in Real Analysis books);

• If $g$ is uniformly continuous, then any restriction of $g$ is uniformly continuous (it follows of definition of uniform continuity).

Now, notice that

• the function $g:[3,5]\to\mathbb{R}$ given by $g(x) = \cfrac{x^2 + 5x - 7}{(x^2 - 9x + 8)(x-2)}$ is continuous;

• your function $f$ is a restriction of $g$.