CIIM 2009 Problem 5

Let $f:\mathbb{R} \to \mathbb{R}$ , such that

i) For all $a \in \mathbb{R}$ and all $\epsilon > 0$ , exists $\delta > 0$ such that $|x-a| < \delta \Rightarrow f(x) < f(a) + \epsilon.$ ii) For all $b\in \mathbb{R}$ and all $\epsilon > 0$ , exists $x,y \in \mathbb{R}$ with $b - \epsilon < x < b < y < b + \epsilon$ , such that $|f(x)-f(b)|< \epsilon$ and $|f(y)-f(b)| < \epsilon.$ Prove that if $f(a) < d < f(d)$ there exists $c$ with $a < c < b$ or $b < c < a$ such that $f(c) = d$ .