1. ## Continuity Problem

1) If g(x) is continuous at c and g(c)>0, show that there is an open interval centered at c on which g(x) is always positive.

It makes sense when I read it, I'm confused how I go about proving it.

2. Hint : let $\epsilon = g(c)/2$ in the definition of continuity at $c$.

3. Originally Posted by Janu42
1) If g(x) is continuous at c and g(c)>0, show that there is an open interval centered at c on which g(x) is always positive.

It makes sense when I read it, I'm confused how I go about proving it.
The definition of continuity. $g(c)\in(0,\infty)$ and since this is an open set there exists some $B_{\delta}(g(c))\subseteq (0,\infty)$ and by continuity there exists some $B_{\delta}(c)$ such that $g(B_{\delta}(c))\subseteq B_{\varepsilon}(g(c))\subseteq(0,\infty)$