Functional Limits and Continuity so here is a problem I have been working on:
Assume g is defined and continuous on all of $\mathbb{R}$.
If $g(x_0) > 0$ for a single point  $x_0  \in \mathbb{R}$, then $g(x)$ is in fact strictly positive for uncountably many points.
So far I've come up with the following:
(I've already shown there exists a bijective function between any interval (a, b), a < b, and R so I first take that for granted.) Since this is true, any interval (a, b) contains uncountably many points. Thus, the interval (0, g(x0)) contains uncountably many points. 
Now, here is my issue: how does one use continuity to show that g(x) must contain this interval rigorously? Intuitively, this conclusion seems obvious.

A possible proof is this one :
Let $\epsilon = \frac{g(x_0)}{2}$, then by continuity, there exist $\delta > 0$ such that $\forall x\in ]x_0-\delta, x_0+\delta[, 0<g(x_0)-\epsilon \leq g(x) \leq g(x_0)+\epsilon$
So $\forall x\in ]x_0-\delta, x_0+\delta[, g(x >0)$ and as there are uncountably points in this interval, we have the resultat
