problem with  continuous functions $f,g:\mathbb{R}\longrightarrow\mathbb{R}$
f,g are continuous functions. $\forall q\in\mathbb{Q}$   $f\left(q\right)\leq g\left(q\right)$
I need to prove that $\forall x\in\mathbb{R}$   $f\left(x\right)\leq g\left(x\right)$

Pick $x\in\mathbb{R}\setminus\mathbb{Q}$.
Suppose $f(x)>g(x)$.  Let $\varepsilon>0$ be less than the difference between $f(x)$ and $g(x)$.  Then there exists $\delta>0$ such that if $w$ differs from $x$ by less than $\delta$, then $f(w)-g(w)$ differs from $f(x)-g(x)$ by less than $\varepsilon$.  But some rational numbers are in the interval $(x-\delta,x+\delta)$, so we have a contradiction.
