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