I want to show that if $\displaystyle f:[a,b]\to\mathbb{R}$ is continuous, $\displaystyle N\subseteq [a,b]$ is countable and $\displaystyle \limsup_{x\to a^+}\frac{f(x)-f(a)}{x-a}\geq 0$ on $\displaystyle [a,b]\setminus N$, then $\displaystyle f(x)\geq f(y)$ for all $\displaystyle x,y\in[a,b]$, $\displaystyle x\geq y$.

Harking back to my days in real analysis I can show the result if we ignore the set N (ie., if the limsup is >=0 on all of [a,b]). I'm just stuck on how to 'cut out N', if you like -- I'm used to there being a measure or integral present to take care of a countable set. Any help there would be great.