Continuity of Function with positive restrictions; Proof Help

Question

If the function g is continuous at the point $c$ and $g(c)$ is positive.

How do you show that there exists a $\delta > 0$ such that $g(x) > 0$ for all $x$ satisfying $|x − c| < \delta$ ?

Answer 1

$g$ is continuous at $c \implies \forall ε>0$ $\exists δ>0:\forall x \in D_g, |x-c|<δ \implies |g(x)-g(c)|<ε$

This means that there is a $δ$ for $ε=\frac{g(c)}{10}$, for example, which is positive as well and makes $g(x)$ positive.