Math Help - Unif. Continuity Again

1. Unif. Continuity Again

If $f,g$ are unif. cont. on an interval $[a,b]$ and $g(x) \not= 0$ for $x \in [a,b]$, then $f/g$ is unif. cont. on $[a,b]$

Being able to get started correctly would be helpful. Thanks.

2. Originally Posted by Math2010
If $f,g$ are unif. cont. on an interval $[a,b]$ and $g(x) \not= 0$ for $x \in [a,b]$, then $f/g$ is unif. cont. on $[a,b]$
The interval $[a,b]$ is closed so any continious function is uniformally continuous.
First prove that $\frac{1}{g}$ is continuous.
Without loss of generally, we can assume that $g$ is positive on $[a,b]$.
Therefore, $\left( {\exists c \in [a,b]} \right)\left( {\forall x \in [a,b]} \right)\left[ {0 < g(c) \leqslant g(x)} \right]$, the low point theorem.
Now we have a bound on $\frac{1}{g}$.